11institutetext: L. Köhldorfer 22institutetext: Acoustics Research Institute,
Austrian Academy of Sciences, 22email<EMAIL_ADDRESS>33institutetext: P. Balazs 44institutetext: Acoustics Research Institute,
Austrian Academy of Sciences, 44email<EMAIL_ADDRESS>55institutetext: P. Casazza 66institutetext: University Of Missouri, Dept. of
Mathematics 66email<EMAIL_ADDRESS>77institutetext: S. Heineken
88institutetext: Instituto de Investigaciones Matemáticas Luis A. Santaló
(UBA-CONICET), 88email<EMAIL_ADDRESS>99institutetext: C. Hollomey
1010institutetext: Acoustics Research Institute, Austrian Academy of Sciences,
1010email<EMAIL_ADDRESS>1111institutetext: P. Morillas
1212institutetext: Instituto de Matemática Aplicada San Luis (UNSL-CONICET),
1212email<EMAIL_ADDRESS>1313institutetext: M. Shamsabadi
1414institutetext: Acoustics Research Institute, Austrian Academy of Sciences,
1414email<EMAIL_ADDRESS>
# A Survey of Fusion Frames in Hilbert Spaces

L. Köhldorfer  P. Balazs  P. Casazza  S. Heineken  C. Hollomey  P. Morillas
M. Shamsabadi

###### Abstract

Fusion frames are a very active area of research today because of their myriad
of applications in pure mathematics, applied mathematics, engineering,
medicine, signal and image processing and much more. They provide a great
flexibility for designing sets of vectors for applications and are therefore
prominent in all these areas, including e.g. mitigating the effects of noise
in a signal or giving robustness to erasures. In this chapter, we present the
fundamentals of fusion frame theory with an emphasis on their delicate
relation to frame theory. The goal here is to provide researchers and students
with an easy entry into this topic. Proofs for fusion frames will be self-
contained and differences between frames and fusion frames are analyzed. In
particular, we focus on the subtleties of fusion frame duality. We also
provide a reproducible research implementation.

##  1 Introduction

Hilbert space frames $(\varphi_{i})_{i\in I}$ are (possibly redundant)
sequences of vectors in a Hilbert space satisfying a Parseval inequality (see
Definition 1). This is a generalization of orthogonal expansions. This subject
was introduced in the setting of nonharmonic Fourier series in 1952 in
duffschaef1 . The topic then became dormant for 30 years until it was brought
back to life in daubgromay86  in the setting of data processing. After this,
the subject advanced rapidly and soon became one of the most active areas of
research in applied mathematics. Frames were originally used in signal and
image processing, and later in sampling theory, data compression, time-
frequency analysis, coding theory, and Fourier series. Today, there are ever
increasing applications of frames to problems in pure and applied mathematics,
computer science, engineering, medicine, and physics with new applications
arising regularly (see CK2 ; ole1 ; W  and the ”Further Study” section).
Redundancy plays a fundamental role in applications of frame theory. First, it
gives much greater flexibility in the design of families of vectors for a
particular problem not possible for linearly independent sets of vectors. In
particular, the advantages of redundancy was shown in phase reconstruction,
where we need to determine a vector from the absolute value of its frame
coefficients BCE ; BBCE . Second, we can mitigate the effects of noise in a
signal and get robustness to erasures by spreading information over a wider
range of vectors. For an introduction to frame theory, we recommend CK2 ; ole1
; W .

One can think of a frame as a family of weighted one-dimensional subspaces of
a Hilbert space. As a consequence, frame theory has its limitations. In
particular, they do not work well in distributed processing where one has to
project a signal onto multidimensional subspaces. This requires fusion frames
for the applications which include wireless sensor networks IB , visual and
hearing systems dist3 , geophones in geophysics CG , distributed processing
cakuli08 , packet encoding B ; BKP ; CKLR , parallel processing CK3 , and much
more. Fusion frames also arise in various theoretical applications including
the Kadison-Singer Problem CT2 , and optimal subspace packings CHST ; CHST2 .
All of these problems require working with a countable family of subspaces of
a Hilbert space with positive weights and satisfying a type of Parseval
inequality (see Definition 2). Fusion frames were introduced in caskut04
where they were called ”frames of subspaces”. In a followup paper cakuli08 ,
the name was changed to ”fusion frames” so they would not be confused with
”frames in subspaces”. Because of their many applications, fusion frame theory
has developed very rapidly over the years (see also CK2 ). Here, we give a new
survey on this topic.

###  1.1 Motivation

The motivation for this chapter is to present the fundamentals of fusion frame
theory with an emphasis on results from frame theory which hold, or do not
hold, for fusion frames. The idea is to help future researchers deal with the
delicate differences between frames and fusion frames accurately. As such,
proofs for fusion frames will be as self-contained as possible. Let us mention
that there are very nice articles caskut04 ; cakuli08 , we can recommend
unrestricted, still we think that there is room for this survey.

Finite frame and fusion frame theory became more and more important recently.
It became apparent that there is a necessity for the implementation of fusion
frame approaches. There are, of course, algorithms, but we provide a
reproducible research approach $-$ integrating the codes in the open source
Toolbox LTFAT soendxxl10 ; ltfatnote030 .

Fusion frames have a natural connection to domain decomposition methods, and
can be used for the solution of operator equations oswald09 . In
xxlcharshaare18  a direct fusion frame formulation is used for that. Here it
can be observed that to use the more general definition of the coefficient
spaces $-$ not restricting to the Hilbert direct sum of the given subspaces
$-$ has advantages, see below. Here we adopt this approach, also used in
Asg_15 .

Many concepts of classical frame theory have been generalized to the setting
of fusion frames. Having a proper notion of dual fusion frames permits us to
have different reconstruction strategies. Also, one wants from a proper
definition to yield duality results similar to those known for classical
frames. The duality concept for fusion frames is more elaborate but therefore
also more interesting than the standard frame setting. This is why we decided
to put some focus on this problem, and summarized results from hemo14 ;
HeinekenMorillas18 ; heimoanza14 .

For this we have also chosen to make properties of operators between Hilbert
spaces explicit, and present the proofs of some results that were often only
considered implicitly.

So, in summary, we provide a new survey on fusion frame theory, aiming to
provide an easier entry point into this theory for mathematicians.

###  1.2 Structure

This chapter is structured as follows: Section 1 introduces and motivates this
chapter. In particular, we present a list of differences between frame and
fusion frame theory (which we present throughout the chapter) in the next
subsection. Section 2 introduces notation, general preliminary results and
some basic facts on frame theory, which we apply on many occasions throughout
these notes. Section 3 presents the notion of fusion frames, Bessel fusion
sequences and their canonically associated frame-related operators. The reader
familiar with frame theory will recognise many parallels between fusion frames
and frames in this section. We also add a subsection on operators between
Hilbert direct sums, which serves as a preparation for later and leads to a
better understanding of many operators occurring in this chapter. In Section
4, we present other notions in fusion frame theory including generalizations
of orthonormal bases, Riesz bases, exact frames and minimal sequences to the
fusion frame setting. In Section 5, we study fusion frames under the action of
bounded operators, where we present a number of differences between frame and
fusion frame theory. In Section 6, we present the theory of dual fusion
frames. Duality theory probably provides the greatest contrast between frames
and fusion frames. We first show the reader how the concept of dual fusion
frame arrived. We study properties, present results and provide examples of
this concept. We discuss particular cases and other approaches. In Section 7
we focus on fusion frames in finite dimensional Hilbert spaces. In Section 8
we present thoughts about concrete implementations in the LTFAT toolbox
soendxxl10 , and a link to a reproducible research page. Finally, in Section 9
we list a collection of topics on fusion frames for further study, which are
not presented in this chapter.

###  1.3 ”There Be Dragons” - traps to be avoided when going from frame theory
to fusion frame theory

In the following, we give an overview of the differences between properties of
frames and properties of fusion frames presented in this chapter.

  * •

_Existence of Parseval fusion frames:_ It is known that for every $m\geq
n\in\mathbb{N}$ there is a $m$-element Parseval frame for any $n$-dimensional
Hilbert space CL  (and the vectors may even be equal norm). In the fusion
frame setting, this is not true (see Remark 1). In particular, applying
$S_{V}^{-1/2}$ to the fusion frame does not yield a Parseval fusion frame in
general (see Section 5). Currently, there are no necessary and sufficient
conditions known for the existence of Parseval fusion frames for arbitrary
dimensions of the subspaces.

  * •

_Coefficient space:_ In contrast to frame theory, there is no natural
coefficient space. It can either be chosen to be the Hilbert direct sum of the
defining subspaces, which is the standard approach in fusion frame theory. But
then the coefficient space is different for each fusion system, and a naive
combination of analysis and synthesis is not possible anymore. Or it can be
chosen to be the Hilbert direct sum of copies of the whole Hilbert space $-$
the approach taken in this survey, but then the analysis operator associated
to a fusion frame $V$ maps $\mathcal{H}$ bijectively onto the subspace
$\mathcal{K}_{V}^{2}$ and not the full coefficient space
$\mathcal{K}_{\mathcal{H}}^{2}$ (see Theorem 4.1).

  * •

Exact systems are not necessarily Riesz, see Proposition 5 or caskut04 .

  * •

Applying a bounded surjective operator on a fusion frame does not give a
fusion frame, in general. Even for orthonormal fusion bases this is not true,
see Section 5.

  * •

_Duality:_

    * –

Duality for fusion frames cannot be defined via
$D_{W}C_{V}=\mathcal{I}_{\mathcal{H}}$ as in the frame case, if one wants
fusion frame duality to be compatible with frame duality, see Subsection 6.9.

    * –

For the dual fusion frame definition, an extra operator $Q$ has to be inserted
between $C_{W}$ and $D_{V}$, see Definition 5 and the extensive discussion in
Section 6.

    * –

Duals of fusion Riesz bases are in general not unique, see Subsection 6.8.

    * –

The fusion frame operator of the canonical dual can differ from the inverse of
the fusion frame operator, see Subsection 6.8.

    * –

The canonical dual of the canonical dual of a fusion frame $V$ is not $V$ in
general, see Subsection 6.8.

For a full treatment of fusion frame duality see Section 6.

  * •

The _frame operator of an orthonormal fusion basis $(V_{i})_{i\in I}$ is not
necessarily the identity_. This is only the case, if the associated weights
are uniformly $1$. In fact, the definition of an orthonormal fusion basis is
independent of weights, whereas we have $S_{V^{1}}=\mathcal{I}_{\mathcal{H}}$
for $V^{1}=(V_{i},1)_{i\in I}$, see Theorem 4.2.

##  2 Preliminaries

Throughout this notes, $\mathcal{H}$ is always a separable Hilbert space and
all considered index sets are countable. For a closed subspace $V$ of
$\mathcal{H}$, $\pi_{V}$ denotes the orthogonal projection onto $V$. The set
of positive integers $\\{1,2,3,\dots\\}$ is denoted by $\mathbb{N}$ and
$\delta_{ij}$ denotes the Kronecker-delta. The cardinality of a set $J$ is
denoted by $|J|$. The domain, kernel and range of an operator $T$ is denoted
by $\mathrm{dom}(T)$, $\mathcal{N}(T)$ and $\mathcal{R}(T)$ respectively.
$\mathcal{I}_{X}$ denotes the identity operator on a given space $X$. The set
of bounded operators between two normed spaces $X$ and $Y$ is denoted by
$\mathcal{B}(X,Y)$ and we set $\mathcal{B}(X):=\mathcal{B}(X,X)$. For an
operator $U\in\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})$
($\mathcal{H}_{1}$, $\mathcal{H}_{2}$ Hilbert spaces) with closed range
$\overline{\mathcal{R}(U)}=\mathcal{R}(U)$, its associated _pseudo-inverse_
ole1n  is denoted by $U^{\dagger}$. The pseudo-inverse of $U$ is the unique
operator $U^{\dagger}\in\mathcal{B}(\mathcal{H}_{2},\mathcal{H}_{1})$,
satisfying the three relations

$\mathcal{N}(U^{\dagger})=\mathcal{R}(U)^{\perp},\,\,\,\,\mathcal{R}(U^{\dagger})=\mathcal{N}(U)^{\perp},\,\,\,\,UU^{\dagger}x=x\quad(x\in\mathcal{R}(U)).$
(1)

Moreover, $UU^{\dagger}=\pi_{\mathcal{R(U)}}$ and
$U^{\dagger}U=\pi_{\mathcal{R(U^{\dagger})}}$. We also note that if $U$ has
closed range, then so does $U^{*}$ and it holds
$(U^{*})^{\dagger}=(U^{\dagger})^{*}$. On $\mathcal{R}(U)$ we explicitly have
$U^{\dagger}=U^{*}(UU^{*})^{-1}$. In case $U$ is bounded and invertible, it
holds $U^{\dagger}=U^{-1}$. We refer to ole1n  for more details on the pseudo-
inverse of a closed range operator between Hilbert spaces.

The following important preliminary results will be applied without further
reference throughout this manuscript:

######  Theorem 2.1 (Density Principle gr01 )

Let $X$ and $Y$ be Banach spaces and let $V$ be a dense subspace of $X$. If
$T:V\longrightarrow Y$ is bounded, then there exists a unique bounded
extension $\tilde{T}:X\longrightarrow Y$ of $T$ satisfying
$\|\tilde{T}\|=\|T\|$.

######  Theorem 2.2 (Neumann’s Theorem ole1n )

Let $X$ be a Banach space. If $T\in\mathcal{B}(X)$ satisfies
$\|\mathcal{I}_{X}-T\|<1$, then $T$ is invertible.

######  Theorem 2.3 (Bounded Inverse Theorem ole1n )

Any bounded and bijective operator between two Banach spaces has a bounded
inverse.

######  Theorem 2.4 (Uniform Boundedness Principle ole1n )

Let $X$ be a Banach space and $Y$ be a normed space. Suppose that
$(T_{n})_{n\in\mathbb{N}}$ is a family of bounded operators
$T_{n}:X\longrightarrow Y$ and assume that $(T_{n})_{n\in\mathbb{N}}$
converges pointwise (as $n\longrightarrow\infty$) to some operator
$T:X\longrightarrow Y$, $Tx:=\lim_{n\longrightarrow\infty}T_{n}x$. Then $T$
defines a bounded linear operator and we have

$\|T\|\leq\liminf_{n\in\mathbb{N}}\|T_{n}\|\leq\sup_{n\in\mathbb{N}}\|T_{n}\|<\infty.$

###  2.1 Frame theory

We begin with a review of frame theory and present a collection of those
definitions and results which are the most important ones for this manuscript.
In the later sections, we will apply these results mostly without further
reference. We refer to ole1n  for the missing proofs and more details on frame
theory.

###### Definition 1

Let $\varphi=(\varphi_{i})_{i\in I}$ be a countable family of vectors in
$\mathcal{H}$. Then $\varphi$ is called

  * •

a _frame_ for $\mathcal{H}$, if there exist positive constants
$A_{\varphi}\leq B_{\varphi}<\infty$, called _frame bounds_ , such that

$\displaystyle A_{\varphi}\|f\|^{2}\leq\sum_{i\in I}|\langle
f,\varphi_{i}\rangle|^{2}\leq B_{\varphi}\|f\|^{2}\qquad(\forall
f\in\mathcal{H}).$ (2)

  * •

a _frame sequence_ , if it is a frame for
$\overline{\text{span}}(\varphi_{i})_{i\in I}$.

  * •

an _$A_{\varphi}$ -tight_ frame (sequence), or simply _tight_ frame
(sequence), if it is a frame (sequence), whose frame bounds $A_{\varphi}$ and
$B_{\varphi}$ in (2) can be chosen to be equal.

  * •

a _Parseval_ frame (sequence), if it is a $1$-tight frame (sequence).

  * •

an _exact_ frame, if it ceases to be a frame whenever one of the frame vectors
$\varphi_{i}$ is removed from $\varphi$.

  * •

a _Bessel sequence_ for $\mathcal{H}$, if the right but not necessarily the
left inequality in (2) is satisfied for some prescribed $B_{\varphi}>0$. In
this case we call $B_{\varphi}$ a _Bessel bound_.

  * •

a _complete_ sequence, if $\overline{\text{span}}(\varphi_{i})_{i\in
I}=\mathcal{H}$.

  * •

a _Riesz basis_ , if it is a complete sequence for which there exist constants
$0<\alpha_{\varphi}\leq\beta_{\varphi}<\infty$, called _Riesz constants_ ,
such that

$\alpha_{\varphi}\sum_{i\in I}|c_{i}|^{2}\leq\bigg{\|}\sum_{i\in
I}c_{i}\varphi_{i}\bigg{\|}^{2}\leq\beta_{\varphi}\sum_{i\in I}|c_{i}|^{2}$
(3)

for all finite scalar sequences $(c_{i})_{i\in I}\in\ell^{00}(I)$.

  * •

a _minimal_ sequence, if for each $i\in I$,
$\varphi_{i}\notin\overline{\text{span}}(\varphi_{k})_{k\in I,k\neq i}$.

To any sequence $\varphi=(\varphi_{i})_{i\in I}$ of vectors in $\mathcal{H}$,
we may associate the following canonical frame-related operators xxlstoeant11
:

  * •

The _synthesis operator_
$D_{\varphi}:\text{dom}(D_{\varphi})\subseteq\ell^{2}(I)\longrightarrow\mathcal{H}$,
where $\text{dom}(D_{\varphi})=\big{\\{}(c_{i})_{i\in
I}\in\ell^{2}(I):\sum_{i\in I}c_{i}\varphi_{i}\in\mathcal{H}\big{\\}}$ and
$D_{\varphi}(c_{i})_{i\in I}=\sum_{i\in I}c_{i}\varphi_{i}$.

  * •

The _analysis operator_
$C_{\varphi}:\text{dom}(C_{\varphi})\subseteq\mathcal{H}\longrightarrow\ell^{2}(I)$,
where $\text{dom}(C_{\varphi})=\\{f\in\mathcal{H}:(\langle
f,\varphi_{i}\rangle)_{i\in I}\in\ell^{2}(I)\\}$ and $C_{\varphi}f=(\langle
f,\varphi_{i}\rangle)_{i\in I}$.

  * •

The _frame operator_
$S_{\varphi}:\text{dom}(S_{\varphi})\subseteq\mathcal{H}\longrightarrow\mathcal{H}$,
where $\text{dom}(S_{\varphi})=\big{\\{}f\in\mathcal{H}:\sum_{i\in I}\langle
f,\varphi_{i}\rangle\varphi_{i}\in\mathcal{H}\big{\\}}$ and
$S_{\varphi}f=\sum_{i\in I}\langle f,\varphi_{i}\rangle\varphi_{i}$.

It can be shown that $\varphi$ is a Bessel sequence with Bessel bound
$B_{\varphi}$ if and only if $\text{dom}(D_{\varphi})=\ell^{2}(I)$ and
$\|D_{\varphi}\|\leq\sqrt{B_{\varphi}}$. In this case,
$D^{*}_{\varphi}=C_{\varphi}\in\mathcal{B}(\mathcal{H},\ell^{2}(I))$. If
$\varphi$ is a frame, then $S_{\varphi}$ is a bounded, self-adjoint, positive
and invertible operator, yielding the possibility of _frame reconstruction_
via
$\mathcal{I}_{\mathcal{H}}=S_{\varphi}S_{\varphi}^{-1}=S_{\varphi}^{-1}S_{\varphi}$,
i.e.

$f=\sum_{i\in I}\langle
f,S_{\varphi}^{-1}\varphi_{i}\rangle\varphi_{i}=\sum_{i\in I}\langle
f,\varphi_{i}\rangle S_{\varphi}^{-1}\varphi_{i}\qquad(\forall
f\in\mathcal{H}).$ (4)

The family $(\widetilde{\varphi_{i}})_{i\in
I}:=(S_{\varphi}^{-1}\varphi_{i})_{i\in I}$ in (4) is again a frame (called
the _canonical dual frame_). More generally, a frame $(\psi_{i})_{i\in I}$,
which satisfies

$f=\sum_{i\in I}\langle f,\varphi_{i}\rangle\psi_{i}=\sum_{i\in I}\langle
f,\psi_{i}\rangle\varphi_{i}\qquad(\forall f\in\mathcal{H}),$ (5)

is called a _dual frame_ of $\varphi$. This means, that frames yield (possibly
non-unique and redundant) series expansions of elements in a separable Hilbert
space similar to orthonormal bases, whereas in stark contrast to the latter,
the frame vectors are not necessarily orthogonal and may be linearly
dependent. Moreover, frames and Bessel sequences can often be characterized by
properties of the frame-related operators. For instance, a frame is
$A_{\varphi}$-tight if and only if
$S_{\varphi}=A_{\varphi}\cdot\mathcal{I}_{\mathcal{H}}$. In this case, frame
reconstruction reduces to $f=A_{\varphi}^{-1}\sum_{i\in I}\langle
f,\varphi_{i}\rangle\varphi_{i}$. Further results in this direction are given
below:

###### Theorem 2.5

ch08  Let $\varphi=(\varphi_{i})_{i\in I}$ be a countable family of vectors in
$\mathcal{H}$. Then the following are equivalent.

  1. (i)

$\varphi$ is a Bessel sequence for $\mathcal{H}$.

  2. (ii)

The synthesis operator $D_{\varphi}$ is well-defined on $\ell^{2}(I)$ and
bounded.

  3. (iii)

The analysis operator $C_{\varphi}$ is well-defined on $\mathcal{H}$ and
bounded.

###### Theorem 2.6

ch08  Let $\varphi=(\varphi_{i})_{i\in I}$ be a countable family of vectors in
$\mathcal{H}$. Then the following are equivalent.

  1. (i)

$\varphi$ is a frame (resp. Riesz basis) for $\mathcal{H}$.

  2. (ii)

The synthesis operator $D_{\varphi}$ is well-defined on $\ell^{2}(I)$, bounded
and surjective (resp. well-defined on $\ell^{2}(I)$, bounded and bijective).

  3. (iii)

The analysis operator $C_{\varphi}$ is well-defined on $\mathcal{H}$, bounded,
injective and has closed range (resp. well-defined on $\mathcal{H}$, bounded
and bijective).

In particular, the latter result implies that every Riesz basis is a frame.
Below we give other equivalent conditions for a frame being a Riesz basis.
Recall, that two families $(\varphi_{i})_{i\in I}$ and $(\psi_{i})_{i\in I}$
of vectors with same index set are called _biorthogonal_ , if
$\langle\varphi_{i},\psi_{j}\rangle=\delta_{ij}$ for all $i,j\in I$.

###### Theorem 2.7

ole1n  Let $\varphi=(\varphi_{i})_{i\in I}$ be a frame for $\mathcal{H}$. Then
the following are equivalent.

  1. (i)

$\varphi$ is a Riesz basis for $\mathcal{H}$.

  2. (ii)

$\varphi$ is minimal.

  3. (iii)

$\varphi$ is exact.

  4. (iv)

$\varphi$ has a biorthogonal sequence.

  5. (iii)

$(\varphi_{i})_{i\in I}$ and $(S_{\varphi}^{-1}\varphi_{i})_{i\in I}$ are
biorthogonal.

  6. (iii)

The canonical dual frame $(S_{\varphi}^{-1}\varphi_{i})_{i\in I}$ is the
unique dual frame of $\varphi$.

Bessel sequences, frames and Riesz bases can be characterized in terms of
operator-perturbations of orthonormal bases, see below.

###### Theorem 2.8

ole1n  Let $\mathcal{H}$ be a separable Hilbert space. Then the following
hold:

  1. (i)

The Bessel sequences for $\mathcal{H}$ are precisely the families of the form
$(Ue_{i})_{i\in I}$, where $U\in\mathcal{B}(\mathcal{H})$ and $(e_{i})_{i\in
I}$ is an orthonormal basis for $\mathcal{H}$.

  2. (ii)

The frames for $\mathcal{H}$ are precisely the families of the form
$(Ue_{i})_{i\in I}$, where $U\in\mathcal{B}(\mathcal{H})$ is surjective and
$(e_{i})_{i\in I}$ is an orthonormal basis for $\mathcal{H}$.

  3. (iii)

The Riesz bases for $\mathcal{H}$ are precisely the families of the form
$(Ue_{i})_{i\in I}$, where $U\in\mathcal{B}(\mathcal{H})$ is bijective and
$(e_{i})_{i\in I}$ is an orthonormal basis for $\mathcal{H}$.

This allows us to formulate a result about which operators preserve certain
properties:

###### Corollary 1

ole1n  Let $\mathcal{H}$ be a separable Hilbert space. Then the following
hold:

  1. (i)

If $(\varphi_{i})_{i\in I}$ is a Bessel sequence for $\mathcal{H}$ and
$U:\mathcal{H}\longrightarrow\mathcal{H}$ is bounded, then
$(U\varphi_{i})_{i\in I}$ is a Bessel sequence for $\mathcal{H}$ as well.

  2. (ii)

If $(\varphi_{i})_{i\in I}$ is a frame for $\mathcal{H}$ and
$U:\mathcal{H}\longrightarrow\mathcal{H}$ is bounded and surjective, then
$(U\varphi_{i})_{i\in I}$ is a frame for $\mathcal{H}$ as well.

  3. (iii)

If $(\varphi_{i})_{i\in I}$ is a Riesz basis for $\mathcal{H}$ and
$U:\mathcal{H}\longrightarrow\mathcal{H}$ is bounded and bijective, then
$(U\varphi_{i})_{i\in I}$ is a Riesz basis for $\mathcal{H}$ as well.

We conclude this section with the following characterization of all dual
frames of a given frame $\varphi$, which stems from a characterization of all
bounded left-inverses of the analysis operator $C_{\varphi}$ (analogous to
Lemma 17) and an application of Theorem 2.8.

###### Theorem 2.9

ole1n  Let $\varphi=(\varphi_{i})_{i\in I}$ be a frame for $\mathcal{H}$. Then
the dual frames of $\varphi$ are precisely the families

$(\psi_{i})_{i\in I}=\Big{(}S_{\varphi}^{-1}\varphi_{i}+h_{i}-\sum_{k\in
I}\langle S_{\varphi}^{-1}\varphi_{i},\varphi_{k}\rangle h_{k}\Big{)}_{i\in
I},$ (6)

where $(h_{i})_{i\in I}$ is a Bessel sequence for $\mathcal{H}$.

By the above result and Theorem 2.7, a frame, which is not a Riesz basis, has
infinitely many dual frames. In case $\varphi$ is a Riesz basis, we see that
the remainder terms $h_{i}-\sum_{k\in I}\langle
S_{\varphi}^{-1}\varphi_{i},\varphi_{k}\rangle h_{k}$ in (6) vanish, since
$(\varphi_{i})_{i\in I}$ and $(S_{\varphi}^{-1}\varphi_{i})_{i\in I}$ are
biorthogonal in this case.

##  3 Fusion frames

In this section, we present the basic definitions and results regarding fusion
frames, Bessel fusion sequences and their relation to the fusion frame-related
operators. Because this is a survey on this topic, we will give full proofs.
The reader will notice many parallels between frames and fusion frames in this
section. However, as we have already mentioned and will see later, fusion
frame theory raises a lot of unexpected and interesting questions, which are
non-existent for Hilbert space frames.

At first, we motivate the concept of fusion frames by generalizing the frame
definition. We rewrite the terms $|\langle f,\varphi_{i}\rangle|^{2}$ from the
frame inequalities (2) as follows:

$\displaystyle|\langle f,\varphi_{i}\rangle|^{2}$ $\displaystyle=\langle
f,\varphi_{i}\rangle\langle\varphi_{i},f\rangle$
$\displaystyle=\big{\langle}\langle
f,\varphi_{i}\rangle\varphi_{i},f\big{\rangle}$
$\displaystyle=\|\varphi_{i}\|^{2}\left\langle\left\langle
f,\frac{\varphi_{i}}{\|\varphi_{i}\|}\right\rangle\frac{\varphi_{i}}{\|\varphi_{i}\|},f\right\rangle.$
(7)

If we set
$V_{i}:=\overline{\text{span}}\\{\varphi_{i}\\}=\text{span}\\{\varphi_{i}\\}$,
then the singleton $\left\\{\|\varphi_{i}\|^{-1}\cdot\varphi_{i}\right\\}$ is
an orthonormal basis for $V_{i}$. In particular, the orthogonal projection
$\pi_{V_{i}}$ of $\mathcal{H}$ onto the one-dimensional subspace $V_{i}$ is
given by $\pi_{V_{i}}f=\left\langle
f,\frac{\varphi_{i}}{\|\varphi_{i}\|}\right\rangle\frac{\varphi_{i}}{\|\varphi_{i}\|}$.
Inserting that into (3) and setting $v_{i}:=\|\varphi_{i}\|$ yields

$\displaystyle|\langle f,\varphi_{i}\rangle|^{2}$
$\displaystyle=v_{i}^{2}\langle\pi_{V_{i}}f,f\rangle$
$\displaystyle=v_{i}^{2}\langle\pi_{V_{i}}^{2}f,f\rangle$
$\displaystyle=v_{i}^{2}\langle\pi_{V_{i}}f,\pi_{V_{i}}f\rangle=v_{i}^{2}\|\pi_{V_{i}}f\|^{2}.$

This allows us to rewrite the frame inequalities to (cakuli08, , Proposition
2.14)

$A_{\varphi}\|f\|^{2}\leq\sum_{i\in I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}\leq
B_{\varphi}\|f\|^{2}\qquad(\forall f\in\mathcal{H}).$

Therefore, we see that frames can also be viewed as weighted sequences of
$1$-dimensional closed subspaces satisfying the inequalities (2). Admitting
arbitrary weights - in the sense of weighted frames xxljpa1  \- and in
particular, arbitrary closed subspaces leads to the definition of a fusion
frame and related notions caskut04 :

###### Definition 2

Let $V=(V_{i},v_{i})_{i\in I}$ be a countable family of closed subspaces
$V_{i}$ of $\mathcal{H}$ and weights $v_{i}>0$. Then $V$ is called

  * •

a _fusion frame_ for $\mathcal{H}$, if there exist positive constants
$0<A_{V}\leq B_{V}<\infty$, called _fusion frame bounds_ , such that

$\displaystyle A_{V}\|f\|^{2}\leq\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}\leq B_{V}\|f\|^{2}\qquad(\forall
f\in\mathcal{H}).$ (8)

  * •

a _fusion frame sequence_ , if it is a fusion frame for
$\mathcal{H}:=\overline{\text{span}}(V_{i})_{i\in I}$.

  * •

an _$A_{V}$ -tight fusion frame (sequence)_, or simply _tight fusion frame
(sequence)_ , if it is a fusion frame (sequence), whose fusion frame bounds
$A_{V}$ and $B_{V}$ in (8) can be chosen to be equal.

  * •

a _Parseval fusion frame (sequence)_ , if it is a $1$-tight fusion frame
(sequence).

  * •

an _exact fusion frame_ , if it is a fusion frame, that ceases to be a fusion
frame whenever one component $(V_{k},v_{k})$ is removed from $V$.

  * •

a _Bessel fusion sequence_ for $\mathcal{H}$, if the right but not necessarily
the left inequality in (8) is satisfied for some $B_{V}>0$. In this case we
call $B_{V}$ a _Bessel fusion bound_.

###### Remark 1

(a) For any Bessel fusion sequence $V=(V_{i},v_{i})_{i\in I}$ (and hence any
fusion frame), we will always implicitly assume (without loss of generality),
that $V_{i}\neq\\{0\\}$ for all $i\in I$, because a zero-space $V_{i}=\\{0\\}$
does not contribute to the sum in (8) anyway. This is done to avoid case
distinctions in some of the theoretical results.

(b) As a first glimpse at the differences between frames and fusion frames,
consider the following: It is known that for every $m,n\in\mathbb{N}$ there is
a $m$-element Parseval frame for any $n$-dimensional Hilbert space CL  (and
the vectors may even be equal norm). The situation for fusion frames is much
more complicated. For example, there is no Parseval fusion frame
$V=(V_{i},v_{i})_{i=1}^{2}$ for $\mathbb{R}^{3}$ consisting of two
2-dimensional subspaces for any choice of weights. This is true since, in this
case, there must exist a unit vector $f$ contained in the $1$-dimensional
space $V_{1}\cap V_{2}$ and so
$v_{1}^{2}\|\pi_{V_{1}}f\|^{2}+v_{2}^{2}\|\pi_{V_{2}}f\|^{2}=(v_{1}^{2}+v_{2}^{2})\|f\|^{2}$,
while a unit vector $f\in V_{1}\setminus V_{2}$ will satisfy
$v_{1}^{2}\|\pi_{V_{1}}f\|^{2}+v_{2}^{2}\|\pi_{V_{2}}f\|^{2}<(v_{1}^{2}+v_{2}^{2})\|f\|^{2}$,
and thus we cannot have $S_{V}=\mathcal{I}_{\mathcal{H}}$ (see Corollary 4).
Currently, there are no necessary and sufficient conditions known for the
existence of Parseval fusion frames for arbitrary dimensions of the subspaces.

(c) A converse to our motivation for fusion frames via frames is the
following: Assume that $V=(V_{i},v_{i})_{i\in I}$ is a fusion frame such that
$\text{dim}(V_{i})=1$ for all $i\in I$. Then, for every $i$, there exists a
suitable vector $\varphi_{i}$, such that
$V_{i}=\text{span}\\{\varphi_{i}\\}=\overline{\text{span}}\\{\varphi_{i}\\}$
and $\pi_{V_{i}}f=\langle
f,\|\varphi_{i}\|^{-1}\cdot\varphi_{i}\rangle\|\varphi_{i}\|^{-1}\cdot\varphi_{i}$.
It is now easy to see, that $V$ being a fusion frame is equivalent to
$(v_{i}\|\varphi_{i}\|^{-1}\cdot\varphi_{i})$ being a frame.

(d) If $V=(V_{i},v_{i})_{i\in I}$ is a fusion frame for $\mathcal{H}$, then
the family $(V_{i})_{i\in I}$ of subspaces $V_{i}$ is _complete_ , i.e. it
holds $\overline{\text{span}}(V_{i})_{i\in I}=\mathcal{H}$. To see this,
assume for the contrary, that there exists a non-zero
$g\in\big{(}\overline{\text{span}}(V_{i})_{i\in I}\big{)}^{\perp}$. Then this
$g$ would violate the lower fusion frame inequality in (8).

A different motivation for fusion frames comes from the following problem:

###### Question 1

Goal Let $(V_{i})_{i\in I}$ be a family of closed subspaces of $\mathcal{H}$
and assume that for every $i\in I$, $(\varphi_{ij})_{j\in J_{i}}$ is a frame
for $V_{i}$. Give necessary and sufficient conditions on the subspaces
$V_{i}$, such that the collection $(\varphi_{ij})_{i\in I,j\in J_{i}}$ of all
local frames (including all resulting repetitions of the vectors
$\varphi_{ij}$) forms a global frame for the whole Hilbert space
$\mathcal{H}$.

The following result shows that the notion of a fusion frame is perfectly
fitting for this task.

###### Theorem 3.1

cakuli08  Let $(V_{i})_{i\in I}$ be a family of closed subspaces of
$\mathcal{H}$ and $(v_{i})_{i\in I}$ be a family of weights. For every $i\in
I$, let $(\varphi_{ij})_{j\in J_{i}}$ be a frame for $V_{i}$ with frame bounds
$A_{i}$ and $B_{i}$, and suppose that $0<A=\inf_{i\in I}A_{i}\leq\sup_{i\in
I}B_{i}=B<\infty$. Then the following are equivalent:

  1. (i)

$(V_{i},v_{i})_{i\in I}$ is a fusion frame for $\mathcal{H}$.

  2. (ii)

$(v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$ is a frame for $\mathcal{H}$.

In particular, if $(V_{i},v_{i})_{i\in I}$ is a fusion frame with bounds
$A_{V}\leq B_{V}$, then $(v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$ is a frame
for $\mathcal{H}$ with frame bounds $AA_{V}\leq BB_{V}$. Conversely, if
$(v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$ is a frame for $\mathcal{H}$ with
bounds $A_{v\varphi}\leq B_{v\varphi}$, then $(V_{i},v_{i})_{i\in I}$ is a
fusion frame with fusion frame bounds bounds
$\frac{A_{v\varphi}}{B}\leq\frac{B_{v\varphi}}{A}$.

###### Proof

Since $(\varphi_{ij})_{j\in J_{i}}$ is a frame for $V_{i}$, we formally see
that

$\displaystyle A\sum_{i\in I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}$
$\displaystyle\leq\sum_{i\in I}v_{i}^{2}A_{i}\|\pi_{V_{i}}f\|^{2}$
$\displaystyle\leq\sum_{i\in I}v_{i}^{2}\sum_{j\in
J_{i}}|\langle\pi_{V_{i}}f,\varphi_{ij}\rangle|^{2}=\sum_{i\in I}\sum_{j\in
J_{i}}|\langle f,v_{i}\varphi_{ij}\rangle|^{2}$ $\displaystyle\leq\sum_{i\in
I}v_{i}^{2}B_{i}\|\pi_{V_{i}}f\|^{2}\leq B\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}.$

Thus, if $(V_{i},v_{i})_{i\in I}$ is a fusion frame for $\mathcal{H}$ with
bounds $A_{V}\leq B_{V}$, then we obtain

$AA_{V}\|f\|^{2}\leq\sum_{i\in I}\sum_{j\in J_{i}}|\langle
f,v_{i}f_{ij}\rangle|^{2}\leq BB_{V}\|f\|^{2}.$

Conversely, if $(v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$ is a frame for
$\mathcal{H}$ with frame bounds $A_{v\varphi}\leq B_{v\varphi}$, then

$\frac{A_{v\varphi}}{B}\|f\|^{2}\leq\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}\leq\frac{B_{v\varphi}}{A}\|f\|^{2}.$

This completes the proof.

The special case $A=A_{i}=K=B_{i}=B$, for every $i\in I$, in the latter result
yields the following corollary.

###### Corollary 2

cakuli08  Let $(V_{i})_{i\in I}$ be a family of closed subspaces of
$\mathcal{H}$ and $(v_{i})_{i\in I}$ be a family of weights. For every $i\in
I$, let $(\varphi_{ij})_{j\in J_{i}}$ be a Parseval frame for $V_{i}$. Then,
for every $K>0$, the following are equivalent:

  1. (i)

$(V_{i},v_{i})_{i\in I}$ is a $K$-tight fusion frame for $\mathcal{H}$.

  2. (ii)

$(v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$ is a $K$-tight frame for
$\mathcal{H}$.

This $-$ corresponding to the approach in many applications that one considers
a fusion frame with local frames for each of its subspaces $-$ leads to the
following definition, see also Sections 6.3 and 8:

###### Definition 3

cakuli08  Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion frame (Bessel fusion
sequence), and let $\varphi^{(i)}:=(\varphi_{ij})_{j\in J_{i}}$ be a frame for
$V_{i}$ for $i\in I$ with frame bounds $A_{i}$ and $B_{i}$. Assume that
$A=\inf A_{i}>0$ and $B=\sup B_{i}<\infty$. Then
$(V_{i},v_{i},\varphi^{(i)})_{i\in I}$ is called a _fusion frame system_
(_Bessel fusion system_).

In the following, we collect a few observations on the weights $v_{i}$
associated to a Bessel fusion sequence. For that, the following notion will be
useful: We call a family $(v_{i})_{i\in I}$ of weights _semi-normalized_ (see
also xxlmult1 ), if there exist positive constants $m\leq M$ such that $m\leq
v_{i}\leq M$ for all $i\in I$.

###### Lemma 1

heimoanza14  Let $V=(V_{i},v_{i})_{i\in I}$ be a Bessel fusion sequence with
Bessel fusion bound $B_{V}$. Then $v_{i}\leq\sqrt{B_{V}}$ for all $i\in I$.

###### Proof

Since we always assume $V_{i}\neq\\{0\\}$ for a Bessel fusion sequence
$(V_{i},v_{i})_{i\in I}$, we may choose some non-zero $f\in V_{i}$ and see
that

$v_{i}^{2}\|f\|^{2}=v_{i}^{2}\left\|\pi_{V_{i}}f\right\|^{2}\leq\sum_{i\in
I}v_{i}^{2}\left\|\pi_{V_{i}}f\right\|^{2}\leq B_{V}\|f\|^{2}$

for every $i\in I$. Dividing by $\|f\|^{2}$ yields the result.

By Lemma 1, the weights $v_{i}$ associated to a fusion frame
$V=(V_{i},v_{i})_{i\in I}$ are uniformly bounded from above by the square root
of its upper frame bound. However, $(v_{i})_{i\in I}$ needs not to be semi-
normalized in general. An easy counter-example is the Parseval fusion frame
$(\mathcal{H},2^{-n})_{n=1}^{\infty}$ for $\mathcal{H}$.

###### Lemma 2

Let $(w_{i})_{i\in I}$ be a semi-normalized family of weights with respect to
the constants $m\leq M$. Then $(V_{i},v_{i})_{i\in I}$ is a fusion frame for
$\mathcal{H}$ if and only if $(V_{i},v_{i}w_{i})_{i\in I}$ is a fusion frame
for $\mathcal{H}$. In particular, $(V_{i},v_{i})_{i\in I}$ is a Bessel fusion
sequence for $\mathcal{H}$ if and only if $(V_{i},v_{i}w_{i})_{i\in I}$ is a
Bessel fusion sequence for $\mathcal{H}$.

###### Proof

Similar to the proof of Theorem 3.1, the statement follows from

$m^{2}\sum_{i\in I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}\leq\sum_{i\in
I}v_{i}^{2}w_{i}^{2}\|\pi_{V_{i}}f\|^{2}\leq M^{2}\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}\qquad(\forall f\in\mathcal{H}).$

Next we introduce the fusion frame-related operators and characterize Bessel
fusion sequences and fusion frames with them.

The canonical representation space for frames is the sequence space
$\ell^{2}(I)$. In the theory of fusion frames we consider generalizations of
$\ell^{2}(I)$. For a family $(V_{i})_{i\in I}$ of Hilbert spaces (e.g. $V_{i}$
closed subspaces of $\mathcal{H}$), the _Hilbert direct sum_
$\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$, also called the _direct
sum of the Hilbert spaces_ $V_{i}$ conw1 ; cas03 , is defined by

$\Big{(}\sum_{i\in I}\oplus V_{i}\Big{)}_{\ell^{2}}=\left\\{(f_{i})_{i\in
I}\in(V_{i})_{i\in I}:\sum_{i\in I}\|f_{i}\|^{2}<\infty\right\\}.$

It is easy to verify that for $(f_{i})_{i\in I},(g_{i})_{i\in
I}\in\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$, the operation

$\langle(f_{i})_{i\in I},(g_{i})_{i\in I}\rangle_{(\sum_{i\in I}\oplus
V_{i})_{\ell^{2}}}:=\sum_{i\in I}\langle f_{i},g_{i}\rangle$

defines an inner product on $\big{(}\sum_{i\in I}\oplus
V_{i}\big{)}_{\ell^{2}}$ (see also conw1 ), yielding the norm

$\|(f_{i})_{i\in I}\|^{2}_{(\sum_{i\in I}\oplus V_{i})_{\ell^{2}}}=\sum_{i\in
I}\|f_{i}\|^{2}.$ (9)

Moreover, adapting any elementary completeness proof for $\ell^{2}(I)$ yields
that the space $\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$ is
complete with respect to the norm (9), i.e. a Hilbert space (see e.g. kohl21
). Note that $\big{(}\sum_{i\in
I}\oplus\mathbb{C}\big{)}_{\ell^{2}}=\ell^{2}(I)$. In analogy to the dense
inclusion $\ell^{00}(I)\subseteq\ell^{2}(I)$, the space of finite vector
sequences

$\Big{(}\sum_{i\in I}\oplus V_{i}\Big{)}_{\ell^{00}}=\Big{\\{}(f_{i})_{i\in
I}\in(V_{i})_{i\in I}:f_{i}\neq 0\,\,\text{for only finitely
many}\,\,i\Big{\\}}$

is a dense subspace of $\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$.
For the remainder of this manuscript we abbreviate

$\mathcal{K}_{V}^{2}:=\Big{(}\sum_{i\in I}\oplus
V_{i}\Big{)}_{\ell^{2}}\quad\text{and}\quad\mathcal{K}_{V}^{00}:=\Big{(}\sum_{i\in
I}\oplus V_{i}\Big{)}_{\ell^{00}}$

and analogously

$\mathcal{K}_{\mathcal{H}}^{2}:=\Big{(}\sum_{i\in
I}\oplus\mathcal{H}\Big{)}_{\ell^{2}}\quad\text{and}\quad\mathcal{K}_{\mathcal{H}}^{00}:=\Big{(}\sum_{i\in
I}\oplus\mathcal{H}\Big{)}_{\ell^{00}}.$

We will continue our discussion on Hilbert direct sums in Subsection 3.1. For
now, we proceed with the definition and basic properties of the fusion frame-
related operators.

Let $V=(V_{i},v_{i})_{i\in I}$ be an arbitrary weighted sequence of closed
subspaces in $\mathcal{H}$. The following canonical operators associated to
$V$ play a central role in fusion frame theory:

  * •

The _synthesis operator_
$D_{V}:\text{dom}(D_{V})\subseteq\mathcal{K}_{\mathcal{H}}^{2}\longrightarrow\mathcal{H}$,
where $\text{dom}(D_{V})=\big{\\{}(f_{i})_{i\in
I}\in\mathcal{K}_{\mathcal{H}}^{2}:\sum_{i\in
I}v_{i}\pi_{V_{i}}f_{i}\,\,\text{converges}\big{\\}}$ and $D_{V}(f_{i})_{i\in
I}=\sum_{i\in I}v_{i}\pi_{V_{i}}f_{i}$,

  * •

The _analysis operator_
$C_{V}:\text{dom}(C_{V})\subseteq\mathcal{H}\longrightarrow\mathcal{K}_{\mathcal{H}}^{2}$,
where $\text{dom}(C_{V})=\big{\\{}f\in\mathcal{H}:(v_{i}\pi_{V_{i}}f)_{i\in
I}\in\mathcal{K}_{\mathcal{H}}^{2}\big{\\}}$ and
$C_{V}f=(v_{i}\pi_{V_{i}}f)_{i\in I}$,

  * •

The _fusion frame operator_
$S_{V}:\text{dom}(S_{V})\subseteq\mathcal{H}\longrightarrow\mathcal{H}$, where
$\text{dom}(S_{V})=\big{\\{}f\in\mathcal{H}:\sum_{i\in
I}v_{i}^{2}\pi_{V_{i}}f\,\,\text{converges}\big{\\}}$ and $S_{V}f=\sum_{i\in
I}v_{i}^{2}\pi_{V_{i}}f$.

The synthesis operator is always densely defined, since
$\mathcal{K}_{\mathcal{H}}^{00}\subseteq\text{dom}(D_{V})$ is dense in
$\mathcal{K}_{\mathcal{H}}^{2}$, and we always have $D_{V}^{*}=C_{V}$ (see
below for the bounded case). On the other hand, one can show petermitraun2021
, that the analysis operator $C_{V}$, in general, is not necessarily densely
defined. If $C_{V}$ is densely defined (or, equivalently, $D_{V}$ is
closable), then $D_{V}\subseteq C_{V}^{*}$ and in this case,
$\overline{D_{V}}=C_{V}^{*}$. Moreover, one can show that always
$S_{V}=D_{V}C_{V}$ and that $S_{V}$ is symmetric on its domain. We refer to
petermitraun2021  for more details on these (possibly unbounded) operators
associated to general subspace sequences. For our purpose it is sufficient to
treat the bounded case only (compare with Theorem 3.2).

###### Lemma 3

caskut04  Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted sequence of closed
subspaces in $\mathcal{H}$. If
$D_{V}\in\mathcal{B}(\mathcal{K}_{\mathcal{H}}^{2},\mathcal{H})$ then
$D_{V}^{*}=C_{V}$; if
$C_{V}\in\mathcal{B}(\mathcal{H},\mathcal{K}_{\mathcal{H}}^{2})$, then
$C_{V}^{*}=D_{V}$; and either of the two latter cases implies that
$S_{V}=D_{V}C_{V}\in\mathcal{B}(\mathcal{H})$ is self-adjoint.

###### Proof

For $g\in\mathcal{H}$ and $(f_{i})_{i\in I}\in\mathcal{K}_{\mathcal{H}}^{2}$
we have

$\displaystyle\big{\langle}D_{V}^{*}g,(f_{i})_{i\in
I}\big{\rangle}_{\mathcal{K}_{\mathcal{H}}^{2}}$
$\displaystyle=\big{\langle}g,D_{V}(f_{i})_{i\in I}\big{\rangle}$
$\displaystyle=\sum_{i\in I}\langle g,v_{i}\pi_{V_{i}}f_{i}\rangle$
$\displaystyle=\sum_{i\in I}\langle v_{i}\pi_{V_{i}}g,f_{i}\rangle$
$\displaystyle=\big{\langle}(v_{i}\pi_{V_{i}}g)_{i\in I},(f_{i})_{i\in
I}\big{\rangle}_{\mathcal{K}_{\mathcal{H}}^{2}}$
$\displaystyle=\big{\langle}C_{V}g,(f_{i})_{i\in
I}\big{\rangle}_{\mathcal{K}_{\mathcal{H}}^{2}}=\big{\langle}g,C_{V}^{*}(f_{i})_{i\in
I}\big{\rangle}_{\mathcal{K}_{\mathcal{H}}^{2}}.$

This implies the first two statements and the rest is clear.

###### Theorem 3.2

caskut04  Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted sequence of closed
subspaces of $\mathcal{H}$. Then the following are equivalent:

  1. (i)

$V$ is a Bessel fusion sequence with Bessel fusion bound $B_{V}$.

  2. (ii)

The synthesis operator $D_{V}$ associated to $V$ is a bounded operator from
$\mathcal{K}_{\mathcal{H}}^{2}$ into $\mathcal{H}$ with
$\|D_{V}\|\leq\sqrt{B_{V}}$.

  3. (iii)

The analysis operator $C_{V}$ associated to $V$ is a bounded operator from
$\mathcal{H}$ into $\mathcal{K}_{\mathcal{H}}^{2}$ with
$\|C_{V}\|\leq\sqrt{B_{V}}$.

  4. (iv)

The fusion frame operator $S_{V}$ associated to $V$ is bounded on
$\mathcal{H}$ with $\|S_{V}\|\leq B_{V}$.

###### Proof

(i) $\Rightarrow$ (ii) Assume that $V$ is a Bessel fusion sequence with Bessel
fusion bound $B_{V}$. Then for any finite vector sequence $h=(h_{i})_{i\in
J}\in\mathcal{K}_{\mathcal{H}}^{00}$ we have

$\displaystyle\Big{\|}\sum_{i\in J}v_{i}\pi_{V_{i}}h_{i}\Big{\|}^{2}$
$\displaystyle=\sup_{\|g\|=1}\Big{|}\Big{\langle}\sum_{i\in
J}v_{i}\pi_{V_{i}}h_{i},g\Big{\rangle}\Big{|}^{2}$
$\displaystyle=\sup_{\|g\|=1}\Big{|}\sum_{i\in J}v_{i}\langle
h_{i},\pi_{V_{i}}g\rangle\Big{|}^{2}$
$\displaystyle\leq\sup_{\|g\|=1}\Big{(}\sum_{i\in
J}v_{i}\|\pi_{V_{i}}g\|\|h_{i}\|\Big{)}^{2}$
$\displaystyle\leq\sup_{\|g\|=1}\Big{(}\sum_{i\in
J}v_{i}^{2}\|\pi_{V_{i}}g\|^{2}\Big{)}\Big{(}\sum_{i\in
J}\|h_{i}\|^{2}\Big{)}\leq B_{V}\|h\|_{\mathcal{K}_{\mathcal{H}}^{2}}^{2}.$

This implies that $D_{V}$ is bounded on $\mathcal{K}_{\mathcal{H}}^{00}$ by
$\sqrt{B_{V}}$. Since $\mathcal{K}_{\mathcal{H}}^{00}$ is a dense subspace of
$\mathcal{K}_{\mathcal{H}}^{2}$, $D_{V}$ extends to a bounded operator
$D_{V}:\mathcal{K}_{\mathcal{H}}^{2}\longrightarrow\mathcal{H}$ with
$\|D_{V}\|\leq\sqrt{B_{V}}$.

(ii) $\Rightarrow$ (iii) $\Rightarrow$ (iv) This follows from Lemma 3.

(iv) $\Rightarrow$ (i) If $S_{V}$ is bounded with $\|S_{V}\|\leq B_{V}$, then
for all $f\in\mathcal{H}$ we have

$\sum_{i\in I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}=\langle
S_{V}f,f\rangle\leq\|S_{V}f\|\|f\|\leq\|S_{V}\|\|f\|^{2}\leq B_{V}\|f\|^{2},$

i.e. $V$ is a Bessel fusion sequence with Bessel fusion bound $B_{V}$.

###### Remark 2

The proof of Theorem 3.2 reveals, that for any Bessel fusion sequence
$V=(V_{i},v_{i})_{i\in I}$, the sum $\sum_{i\in I}v_{i}^{2}\pi_{V_{i}}f_{i}$
converges unconditionally in $\mathcal{H}$ for all $(f_{i})_{i\in
I}\in\mathcal{K}^{2}_{\mathcal{H}}$.

By Theorem 3.2, the fusion frame operator $S_{V}$ is bounded on $\mathcal{H}$,
whenever $V=(V_{i},v_{i})_{i\in I}$ is a Bessel fusion sequence. If, in
addition, $V$ is a fusion frame, then $S_{V}$ is invertible and thus yields
the possibility of perfect reconstruction, see Theorem 3.3 below.

In analogy to frame theory, fusion frames enable perfect reconstruction
without the assumption, that the subspaces $V_{i}$ are orthogonal. This can be
seen as the most important advantage of fusion frames in comparison with
orthogonal subspace decompositions.

###### Theorem 3.3

caskut04  Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion frame with fusion frame
bounds $A_{V}\leq B_{V}$. Then the fusion frame operator $S_{V}$ is bounded,
self-adjoint, positive and invertible with

$A_{V}\leq\|S_{V}\|\leq B_{V}\quad\text{and}\quad
B_{V}^{-1}\leq\|S_{V}^{-1}\|\leq A_{V}^{-1}.$ (10)

In particular

$f=\sum_{i\in I}v_{i}^{2}\pi_{V_{i}}S_{V}^{-1}f=\sum_{i\in
I}v_{i}^{2}S_{V}^{-1}\pi_{V_{i}}f\qquad(\forall f\in\mathcal{H})$ (11)

where both series converge unconditionally.

###### Proof

If $V$ is a fusion frame, then $V$ in particular is a Bessel fusion sequence.
Hence, Theorem 3.2 and Lemma 3 imply, that $S_{V}$ is bounded and self-
adjoint. Moreover, by the definition of $S_{V}$, we may rewrite the fusion
frame inequalities (8) to

$\langle A_{V}f,f\rangle\leq\langle S_{V}f,f\rangle\leq\langle
B_{V}f,f\rangle\qquad(\forall f\in\mathcal{H}).$ (12)

Thus $S_{V}$ is positive. Furthermore, by manipulating (12), we obtain

$0\leq\big{\langle}(\mathcal{I}_{\mathcal{H}}-B_{V}^{-1}S_{V})f,f\big{\rangle}\leq\Big{\langle}\frac{B_{V}-A_{V}}{B_{V}}f,f\Big{\rangle}\qquad(\forall
f\in\mathcal{H})$

and thus, since $\mathcal{I}_{\mathcal{H}}-B_{V}^{-1}S_{V}$ is self-adjoint,

$\|\mathcal{I}_{\mathcal{H}}-B_{V}^{-1}S_{V}\|=\sup_{\|f\|=1}\langle(\mathcal{I}_{\mathcal{H}}-B_{V}^{-1}S_{V})f,f\rangle\leq\frac{B_{V}-A_{V}}{B_{V}}<1.$

Now, an application of Neumann’s Theorem yields that $B_{V}^{-1}S_{V}$ is
invertible, hence $S_{V}$ is invertible too. In particular, for all
$f\in\mathcal{H}$, we have $f=S_{V}S_{V}^{-1}f=S_{V}^{-1}S_{V}f$, which
implies (11) with unconditional convergence (see Remark 2). Finally, the left-
hand side of (10) follows immediately from (12). To show the right-hand side
of (10), observe that we have $\|S_{V}^{-1}\|\geq\|S_{V}\|^{-1}\geq
B_{V}^{-1}$ and that $A_{V}\|S_{V}^{-1}f\|^{2}\leq\langle
S_{V}S_{V}^{-1}f,S_{V}^{-1}f\rangle=\langle
f,S_{V}^{-1}f\rangle\leq\|S_{V}^{-1}\|\|f\|^{2}$ implies $\|S_{V}^{-1}\|\leq
A_{V}^{-1}$.

So, for any fusion frame $V$, we have the possibility of perfect
reconstruction of any $f\in\mathcal{H}$ via (11), where the subspaces $V_{i}$
need not to be orthogonal and can have non-trivial intersection. In
particular, we can combine fusion frame reconstruction with local frame
reconstruction in a fusion frame system. More precisely, given a fusion frame
system $(V_{i},v_{i},\varphi^{(i)})_{i\in I}$ with associated global frame
$v\varphi$, we obtain several possibilities to perform perfect reconstruction
via the frame vectors $v_{i}\varphi_{ij}$ (see Definition 3), which
demonstrates that fusion frames are tremendously useful for distributed
processing tasks cakuli08 .

###### Lemma 4

cakuli08  Let $V=(V_{i},v_{i},\varphi^{(i)})_{i\in I}$ be a fusion frame
system in $\mathcal{H}$ with global frame $v\varphi$. If
$(\varphi_{ij}^{d})_{j\in J_{i}}$ is a dual of $\varphi^{(i)}$ for every $i\in
I$, then

  1. (i)

$(S_{V}^{-1}v_{i}\varphi^{d}_{ij})_{i\in I,j\in J_{i}}$ is a dual frame of the
global frame $v\varphi$. In particular

$f=\sum_{i\in I}\sum_{j\in
J_{i}}\left<f,v_{i}\varphi_{ij}\right>S_{V}^{-1}v_{i}\varphi_{ij}^{d}\qquad(\forall
f\in\mathcal{H}).$

  2. (ii)

$S_{V}=D_{v\varphi^{d}}C_{v\varphi}=D_{v\varphi}C_{v\varphi^{d}}$, where
$v\varphi^{d}=(v_{i}\varphi_{ij}^{d})_{i\in I,j\in J_{i}}$.

###### Proof

(i) For every $f\in\mathcal{H}$ it holds $\pi_{V_{i}}f=\sum_{i\in
I}\left<f,\varphi_{ij}\right>\varphi_{ij}^{d}$ and hence

$f=\sum_{i\in I}v_{i}^{2}S_{V}^{-1}\pi_{V_{i}}f=\sum_{i\in I}\sum_{j\in
J_{i}}\left<f,v_{i}\varphi_{ij}\right>S_{V}^{-1}v_{i}\varphi_{ij}^{d}.$

(ii) Similarly, we see that

$S_{V}f=\sum_{i\in I}v_{i}^{2}\pi_{V_{i}}f=\sum_{i\in I}\sum_{j\in
J_{i}}\left<f,v_{i}\varphi_{ij}\right>v_{i}\varphi_{ij}^{d},$

where we note that $v\varphi^{d}$ is indeed a frame for $\mathcal{H}$ by
Theorem 3.1.

In view of applications, knowing the action of the inverse fusion frame
operator $S_{V}^{-1}$ on elements $f\in\mathcal{H}$ is of utmost importance.
If one prefers to avoid the inversion of $S_{V}$, approximation procedures
such as the _fusion frame algorithm_ (see Lemma 22) can be performed. However,
for tight fusion frames, the computation of $S_{V}^{-1}$ becomes fairly simple
and thus fusion frame reconstruction can be computed with ease, see (13) and
(14).

First, we note the following reformulation of the inequalities (12).

###### Corollary 3

cakuli08  Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted sequence of closed
subspaces of $\mathcal{H}$ and let $0<A_{V}\leq B_{V}<\infty$. Then the
following are equivalent:

  1. (i)

$V$ is a fusion frame with fusion frame bounds $A_{V}\leq B_{V}$.

  2. (ii)

$A_{V}\cdot\mathcal{I}_{\mathcal{H}}\leq S_{V}\leq
B_{V}\cdot\mathcal{I}_{\mathcal{H}}$ (in the sense of positive operators).

###### Corollary 4

cakuli08  Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted sequence of closed
subspaces of $\mathcal{H}$. Then the following hold.

  1. (i)

$V$ is an $A_{V}$-tight fusion frame if and only if
$S_{V}=A_{V}\cdot\mathcal{I}_{\mathcal{H}}$.

  2. (ii)

$V$ is a Parseval fusion frame if and only if
$S_{V}=\mathcal{I}_{\mathcal{H}}$.

In particular, for an $A_{V}$-tight fusion frame we have

$f=\frac{1}{A_{V}}\sum_{i\in I}v_{i}^{2}\pi_{V_{i}}f\qquad(\forall
f\in\mathcal{H}),$ (13)

and for a Parseval fusion frame

$f=\sum_{i\in I}v_{i}^{2}\pi_{V_{i}}f\qquad(\forall f\in\mathcal{H}).$ (14)

Similar to Theorem 3.2, we can characterize fusion frames in terms of their
associated canonical operators. The crucial observation for this
characterization is

$\|C_{V}f\|_{\mathcal{K}_{\mathcal{H}}^{2}}^{2}=\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}.$ (15)

###### Theorem 3.4

cakuli08  Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted sequence of closed
subspaces of $\mathcal{H}$. Then the following are equivalent:

  1. (i)

$V$ is a fusion frame for $\mathcal{H}$.

  2. (ii)

The synthesis operator $D_{V}$ is well-defined on
$\mathcal{K}^{2}_{\mathcal{H}}$, bounded and surjective.

  3. (iii)

The analysis operator $C_{V}$ is well-defined on $\mathcal{H}$, bounded,
injective and has closed range.

###### Proof

The equivalence (ii) $\Leftrightarrow$ (iii) is generally true for a bounded
operator between Hilbert spaces and its adjoint conw1 . All conditions imply
boundedness, as we will see in the following.

(i) $\Rightarrow$ (ii) If $V$ is a fusion frame, then
$D_{V}\in\mathcal{B}(\mathcal{K}_{\mathcal{H}}^{2},\mathcal{H})$ by Theorem
3.2. If $D_{V}$ was not surjective, then there would exist some non-zero
$h\in\mathcal{R}(D_{V})^{\perp}=\mathcal{N}(D_{V}^{*})=\mathcal{N}(C_{V})$.
However, inserting this $h$ into equation (15) and the lower fusion frame
inequality (8) would lead to a contradiction.

(iii) $\Rightarrow$ (i) If (ii) and (iii) hold, then $C_{V}$ is bounded below
on $\mathcal{H}$ conw1  and also bounded on $\mathcal{H}$ (by Lemma 3). This
means that the fusion frame inequalities are satisfied for all
$f\in\mathcal{H}$.

As we have seen above, the frame-related operators of fusion frames share many
properties with the frame-related operators associated to frames (see Theorem
2.6). Analogously as in classical frame theory, there are $-$ of course $-$
other characterizations of weighted families of closed subspaces being a
fusion frame via these operators. For instance, the following result gives a
characterization of fusion frames and their frame bounds via their synthesis
operator.

###### Proposition 1

HeinekenMorillas18  Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted family of
closed subspaces in $\mathcal{H}$. Then $V$ is a fusion frame for
$\mathcal{H}$ with bounds $A_{V}\leq B_{V}$ if and only if the following two
conditions are satisfied:

  1. (i)

$(V_{i})_{i\in I}$ is complete, i.e. $\overline{\text{span}}(V_{i})_{i\in
I}=\mathcal{H}$.

  2. (ii)

The synthesis operator $D_{V}$ is well defined on
$\mathcal{K}_{\mathcal{H}}^{2}$ and

$A_{V}\|(f_{i})_{i\in I}\|^{2}\leq\|D_{V}(f_{i})_{i\in I}\|^{2}\leq
B_{V}\|(f_{i})_{i\in I}\|^{2}\quad(\forall(f_{i})_{i\in
I}\in\mathcal{N}(D_{V})^{\perp}).$ (16)

###### Proof

First assume that $V$ is a fusion frame with fusion frame bounds $A_{V}\leq
B_{V}$. By Remark 1 (d), $(V_{i})_{i\in I}$ is complete and Theorem 3.4
implies that the upper inequality in (16) is satisfied and that
$\mathcal{R}(C_{V})$ is closed, since $\mathcal{R}(D_{V})$ is closed. The
latter implies that
$\mathcal{N}(D_{V})^{\perp}=\overline{\mathcal{R}(C_{V})}=\mathcal{R}(C_{V})$,
i.e. $\mathcal{N}(D_{V})^{\perp}$ consists all of families of the form
$(v_{i}\pi_{V_{i}}f)_{i\in I}$, where $f\in\mathcal{H}$. Now, for arbitrary
$f\in\mathcal{H}$, we see that

$\Big{(}\sum_{i\in I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}\Big{)}^{2}=|\langle
S_{V}f,f\rangle|^{2}\leq\|S_{V}f\|^{2}\|f\|^{2}\leq\|S_{V}f\|^{2}\frac{1}{A_{V}}\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}.$

This implies that

$A_{V}\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2}\leq\|S_{V}f\|^{2}=\|D_{V}(v_{i}\pi_{V_{i}}f)_{i\in
I}\|^{2},$

as desired.

Conversely, assume that (i) and (ii) are satisfied. Since
$\mathcal{K}_{\mathcal{H}}^{2}=\mathcal{N}(D_{V})\oplus\mathcal{N}(D_{V})^{\perp}$,
(ii) implies that the synthesis operator $D_{V}$ is norm-bounded on
$\mathcal{K}_{\mathcal{H}}^{2}$ by $B_{V}$. Thus, by Theorem 3.2, $V$ is a
Bessel fusion sequence with Bessel fusion bound $B_{V}$ and it only remains to
show the lower fusion frame inequality with respect to $A_{V}$. We first show
that $\mathcal{R}(D_{V})=\mathcal{H}$. By the validity of (i) and
$\text{span}(V_{i})_{i\in I}\subseteq\mathcal{R}(D_{V})\subseteq\mathcal{H}$,
the claim follows if we show that $\mathcal{R}(D_{V})$ is closed. To this end,
let $(g_{n})_{n=1}^{\infty}$ be a Cauchy sequence in $\mathcal{R}(D_{V})$.
Note that $(g_{n})_{n=1}^{\infty}=(D_{V}h_{n})_{n=1}^{\infty}$, where
$(h_{n})_{n=1}^{\infty}$ is a sequence in $\mathcal{N}(D_{V})^{\perp}$, and
the lower inequality in (16) implies that $(h_{n})_{n=1}^{\infty}$ is a Cauchy
sequence in $\mathcal{N}(D_{V})^{\perp}$. Since $\mathcal{N}(D_{V})^{\perp}$
is a closed subspace of $\mathcal{K}_{\mathcal{H}}^{2}$,
$(h_{n})_{n=1}^{\infty}$ converges to some $h\in\mathcal{N}(D_{V})^{\perp}$.
By continuity of $D_{V}$, this means that $(g_{n})_{n=1}^{\infty}$ converges
to $D_{V}h$ in $\mathcal{R}(D_{V})$. Thus $\mathcal{R}(D_{V})$ is closed and
$\mathcal{R}(D_{V})=\mathcal{H}$ follows. Note that due to Theorem 3.4 we can
already deduce that $V$ is a fusion frame. In order to establish that $A_{V}$
is a lower fusion frame bound, recall from (1) that
$D_{V}^{\dagger}D_{V}=\pi_{\mathcal{N}(D_{V})^{\perp}}$ and
$D_{V}D_{V}^{\dagger}=\pi_{\mathcal{R}(D_{V})}=\mathcal{I}_{\mathcal{H}}$.
Hence, (ii) implies that for any $(f_{i})_{i\in
I}\in\mathcal{K}_{\mathcal{H}}^{2}$ we can estimate

$A\|D_{V}^{\dagger}D_{V}(f_{i})_{i\in
I}\|^{2}\leq\|D_{V}D_{V}^{\dagger}D_{V}(f_{i})_{i\in
I}\|^{2}=\|D_{V}(f_{i})_{i\in I}\|^{2}.$

Since also $\mathcal{N}(D_{V}^{\dagger})=\mathcal{R}(D_{V})^{\perp}=\\{0\\}$,
the latter implies that $\|D_{V}^{\dagger}\|^{2}\leq A_{V}^{-1}$. In
particular, this yields
$\|(D_{V}^{*})^{\dagger}\|^{2}=\|C_{V}^{\dagger}\|^{2}\leq A_{V}^{-1}$. Since
$C_{V}^{\dagger}C_{V}$ is the orthogonal projection onto
$\mathcal{R}(C_{V}^{\dagger})=\mathcal{R}((D_{V}^{\dagger})^{*})=\mathcal{N}(D_{V}^{\dagger})^{\perp}=\mathcal{R}(D_{V})=\mathcal{H}$,
we finally see that for arbitrary $f\in\mathcal{H}$ it holds

$\|f\|^{2}=\|C_{V}^{\dagger}C_{V}f\|^{2}\leq\frac{1}{A_{V}}\|C_{V}f\|^{2}=\frac{1}{A_{V}}\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}f\|^{2},$

i.e. $A_{V}$ is a lower fusion frame bound for $V$.

###  3.1 Operators between Hilbert direct sums

Before we proceed with the next section, we provide an excursion into the
realm of linear operators between Hilbert direct sums and their matrix
representations. The results presented here mainly serve as a preparation for
several proof details in the subsequent sections. Thus, the reader who is only
interested in the main results for fusion frames, may skip this subsection.

Instead of considering spaces of the type
$\mathcal{K}_{V}^{2}=\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$,
where each $V_{i}$ is a closed subspace of a given Hilbert space
$\mathcal{H}$, we may consider the slightly more general case, where $V_{i}$
are arbitrary Hilbert spaces, and set

$\displaystyle\Big{(}\sum_{i\in I}\oplus
V_{i}\Big{)}_{\ell^{2}}:=\Big{\\{}(f_{i})_{i\in I}:\,f_{i}\in V_{i},\sum_{i\in
I}\|f_{i}\|_{V_{i}}^{2}<\infty\Big{\\}}.$ (17)

Again, for $f=(f_{i})_{i\in I}$, $g=(g_{i})_{i\in I}\in(\sum_{i\in I}\oplus
V_{i})_{\ell^{2}}$, the operation

$\langle f,g\rangle_{(\sum_{i\in I}\oplus V_{i})_{\ell^{2}}}:=\sum_{i\in
I}\langle f_{i},g_{i}\rangle_{V_{i}}$ (18)

defines an inner product with induced norm

$\|f\|_{(\sum_{i\in I}\oplus V_{i})_{\ell^{2}}}=\big{(}\sum_{i\in
I}\|f_{i}\|_{V_{i}}^{2}\big{)}^{1/2}$

and $(\sum_{i\in I}\oplus V_{i})_{\ell^{2}}$ is complete with respect to this
norm.

Of course, definition (17) extends to the case where the spaces $V_{i}$s are
only pre-Hilbert spaces, resulting in $(\sum_{i\in I}\oplus V_{i})_{\ell^{2}}$
equipped with the inner product (18) being only a pre-Hilbert space. If and
only if all $V_{i}$s are Hilbert spaces, the space $(\sum_{i\in I}\oplus
V_{i})_{\ell^{2}}$ is a Hilbert space, in which case we call it a _Hilbert
direct sum_ of the $(V_{i})_{i\in I}$. In particular, since subspaces of
Hilbert spaces are closed if and only if they are complete, we obtain the
following corollary.

###### Lemma 5

koeba23  Consider the Hilbert direct sum $\big{(}\sum_{i\in I}\oplus
V_{i}\big{)}_{\ell^{2}}$ and for each $i\in I$, let $U_{i}$ be a subspace of
$V_{i}$. Then $\big{(}\sum_{i\in I}\oplus U_{i}\big{)}_{\ell^{2}}$ is a closed
subspace of $\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$ if and only
if $U_{i}$ is a closed subspace of $V_{i}$ for every $i\in I$.

Reminiscent of the previous result, we can prove the following result on
orthogonal complements, see also koeba23 .

###### Lemma 6

koeba23  Consider the Hilbert direct sum $\big{(}\sum_{i\in I}\oplus
V_{i}\big{)}_{\ell^{2}}$ and the (not necessarily closed) subspaces $U_{i}$ of
$V_{i}$ ($i\in I$) and $\big{(}\sum_{i\in I}\oplus U_{i}\big{)}_{\ell^{2}}$ of
$\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$. Then

$\displaystyle\bigg{(}\big{(}\sum_{i\in I}\oplus
U_{i}\big{)}_{\ell^{2}}\bigg{)}^{\perp}=\big{(}\sum_{i\in I}\oplus
U_{i}^{\perp}\big{)}_{\ell^{2}}.$

###### Proof

To show the ”$\supseteq$”-part, choose $g=(g_{i})_{i\in I}\in\big{(}\sum_{i\in
I}\oplus U_{i}^{\perp}\big{)}_{\ell^{2}}$ and observe that for any
$f=(f_{i})_{i\in I}\in\big{(}\sum_{i\in I}\oplus U_{i}\big{)}_{\ell^{2}}$ we
have $\langle f,g\rangle_{(\sum_{i\in I}\oplus V_{i})_{\ell^{2}}}=\sum_{i\in
I}\langle f_{i},g_{i}\rangle_{V_{i}}=0$, implying $g\in\left((\sum_{i\in
I}\oplus U_{i})_{\ell^{2}}\right)^{\perp}$.

To show the ”$\subseteq$”-part, we have to show that if $h=(h_{i})_{i\in
I}\in\big{(}(\sum_{i\in I}\oplus U_{i})_{\ell^{2}}\big{)}^{\perp}$ then
$h_{i}\in U_{i}^{\perp}$ for every $i\in I$. Since $\langle
f,h\rangle_{(\sum_{i\in I}\oplus V_{i})_{\ell^{2}}}=0$ holds for all
$f\in\big{(}\sum_{i\in I}\oplus U_{i}\big{)}_{\ell^{2}}$, this is particularly
true for $f=(...,0,0,f_{i},0,0,...)\in\big{(}\sum_{i\in I}\oplus
U_{i}\big{)}_{\ell^{2}}$ (where $f_{i}\in U_{i}$ is in the $i$-th entry) and
hence we see that in this case $0=\langle f,h\rangle_{(\sum_{i\in I}\oplus
V_{i})_{\ell^{2}}}=\langle f_{i},h_{i}\rangle_{V_{i}}$, which implies
$h_{i}\in U_{i}^{\perp}$. Since this holds for all $i\in I$, the proof is
finished.

Next, we show how orthonormal bases for Hilbert direct sums can naturally be
constructed from local orthonormal bases:

###### Lemma 7

kohl21  For every $i\in I$, let $(e_{ij})_{j\in J_{i}}$ be an orthonormal
basis for $V_{i}$. Set

$\tilde{e}_{ij}=(\delta_{ik}e_{kj})_{k\in
I}=(...,0,0,e_{ij},0,0,...)\,\,\text{($e_{ij}$ in the $i$-th component)}.$

Then $(\tilde{e}_{ij})_{i\in I,j\in J_{i}}$ is an orthonormal basis for
$\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$.

###### Proof

It suffices to show that $(\tilde{e}_{ij})_{i\in I,j\in J_{i}}$ is a complete
orthonormal system. Take two arbitrary elemts
$\tilde{e}_{ij},\tilde{e}_{i^{\prime}j^{\prime}}$: In case $i\neq i^{\prime}$
we immediately see that
$\langle\tilde{e}_{ij},\tilde{e}_{i^{\prime}j^{\prime}}\rangle_{(\sum_{i\in
I}\oplus V_{i})_{\ell^{2}}}=0$ by the definition of
$\langle.,.\rangle_{(\sum_{i\in I}\oplus V_{i})_{\ell^{2}}}$ and the elements
$\tilde{e}_{ij}$. In case $i=i^{\prime}$ we have
$\langle\tilde{e}_{ij},\tilde{e}_{ij^{\prime}}\rangle_{(\sum_{i\in I}\oplus
V_{i})_{\ell^{2}}}=\langle
e_{ij},e_{ij^{\prime}}\rangle_{V_{i}}=\delta_{jj^{\prime}}$. Thus
$(\tilde{e}_{ij})_{i\in I,j\in J_{i}}$ is an orthonormal system. To see that
it is complete, observe that for any $(f_{i})_{i\in I}\in\big{(}\sum_{i\in
I}\oplus V_{i}\big{)}_{\ell^{2}}$ we have

$(f_{i})_{i\in I}=\sum_{i\in I}(\delta_{ik}f_{k})_{k\in I}=\sum_{i\in
I}\Big{(}\delta_{ik}\sum_{j\in J_{k}}\langle f_{k},e_{kj}\rangle
e_{kj}\Big{)}_{k\in I}=\sum_{i\in I}\sum_{j\in J_{i}}\langle
f_{i},e_{ij}\rangle\tilde{e}_{ij}$

and the proof is complete.

Recall the notation

$\mathcal{K}_{V}^{2}:=\Big{(}\sum_{j\in J}\oplus
V_{j}\Big{)}_{\ell^{2}}\quad\text{and}\quad\mathcal{K}_{W}^{2}:=\Big{(}\sum_{i\in
I}\oplus W_{i}\Big{)}_{\ell^{2}},$

where $(V_{j})_{j\in J}$ and $(W_{i})_{i\in I}$ are families of arbitrary
Hilbert spaces associated to the (possibly different) index sets $I$ and $J$.
Assume that for all $i\in I$, $j\in J$ we are given linear operators
$Q_{ij}:\text{dom}(Q_{ij})\subseteq V_{j}\longrightarrow W_{i}$, and let
$f=(f_{j})_{j\in J}\in(\text{dom}(Q_{ij}))_{j\in J}\subseteq(V_{j})_{j\in J}$.
Then

$Qf:=\Big{(}\sum_{j\in J}Q_{ij}f_{j}\Big{)}_{i\in I}$ (19)

defines a linear operator
$Q:\text{dom}(Q)\subseteq\mathcal{K}_{V}^{2}\longrightarrow\mathcal{K}_{W}^{2}$,
where

$\text{dom}(Q)=\Big{\\{}(f_{j})_{j\in J}\in(\text{dom}(Q_{ij}))_{j\in
J}:\sum_{i\in I}\Big{\|}\sum_{j\in
J}Q_{ij}f_{j}\Big{\|}_{W_{i}}^{2}<\infty\Big{\\}}.$ (20)

If we view elements $f=(f_{j})_{j\in J}$ from $\mathcal{K}_{V}^{2}$ as
(possibly infinite) column vectors, then (by assuming $I=J=\mathbb{N}$ without
loss of generality) we can represent the operator $Q$ by the matrix

$Q=\begin{bmatrix}Q_{11}&Q_{12}&Q_{13}&\dots\\\ Q_{21}&Q_{22}&Q_{23}&\dots\\\
Q_{31}&Q_{32}&Q_{33}&\dots\\\ \vdots&\vdots&\vdots&\ddots&\\\ \end{bmatrix},$
(21)

because (19) precisely corresponds to the formal matrix multiplication of $Q$
with $f$.

Our goal is to show that there is a one-to-one correspondence between all
bounded operators $Q\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$
and all matrices $[Q_{ij}]_{j\in J,i\in I}$ of bounded operators
$Q_{ij}\in\mathcal{B}(V_{j},W_{i})$ with $\text{dom}\big{(}[Q_{ij}]_{j\in
J,i\in I}\big{)}=\mathcal{K}_{V}^{2}$ (where the domain is defined as in 20).

For that purpose, we define the coordinate functions

$\mathcal{C}_{V,k}:\mathcal{K}_{V}^{2}\longrightarrow
V_{k},\qquad\mathcal{C}_{V,k}(f_{j})_{j\in J}:=f_{k}\qquad(k\in J).$

Clearly, each operator $\mathcal{C}_{V,k}$ is norm bounded by $1$ and hence
possesses a uniquely determined adjoint operator
$\mathcal{C}_{V,k}^{*}:V_{k}\longrightarrow\mathcal{K}_{V}^{2}$. For $f_{k}\in
V_{k}$ and $(g_{j})_{j\in J}\in\mathcal{K}_{V}^{2}$, the computation

$\displaystyle\big{\langle}\mathcal{C}_{V,k}^{*}f_{k},(g_{j})_{j\in
J}\rangle_{\mathcal{K}_{V}^{2}}$ $\displaystyle=\langle
f_{k},\mathcal{C}_{V,k}(g_{j})_{j\in J}\rangle_{V_{k}}$ $\displaystyle=\langle
f_{k},g_{k}\rangle_{V_{k}}$
$\displaystyle=\langle(...,0,0,f_{k},0,0,...),(g_{j})_{j\in
J}\rangle_{\mathcal{K}_{V}^{2}}$

shows that $\mathcal{C}_{V,k}^{*}$ maps $f_{k}\in V_{k}$ onto
$(...,0,0,f_{k},0,0,...)\in\mathcal{K}_{V}^{2}$ ($f_{k}$ in the $k$-th
component). Analogously, we define

$\mathcal{C}_{W,l}:\mathcal{K}_{W}^{2}\longrightarrow
W_{l},\qquad\mathcal{C}_{W,l}(g_{i})_{i\in I}:=g_{l}\qquad(l\in I)$

with $\|\mathcal{C}_{W,l}\|=1$ and
$\mathcal{C}_{W,l}^{*}:W_{l}\longrightarrow\mathcal{K}_{W}^{2}$ given by
$\mathcal{C}_{W,l}^{*}g_{l}=(0,...,0,g_{l},0,0,...)$ ($g_{l}$ in the $l$-th
component).

###### Proposition 2

kohl21  Let $Q\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$ be
bounded. Then there exists a uniquely determined matrix $[Q_{ij}]_{j\in J,i\in
I}$ of bounded operators $Q_{ij}\in\mathcal{B}(V_{j},W_{i})$, such that the
action of $Q$ on any $(f_{j})_{j\in J}\in\mathcal{K}_{V}^{2}$ is given by the
formal matrix multiplication of $(Q_{ij})_{j\in J,i\in I}$ with $(f_{j})_{j\in
J}$, i.e.

$Q(f_{j})_{j\in J}=\Big{(}\sum_{j\in J}Q_{ij}f_{j}\Big{)}_{i\in I}.$

###### Proof

For each $f=(f_{j})_{j\in J}\in\mathcal{K}_{V}^{2}$ we have $f=\sum_{j\in
J}\mathcal{C}_{V,j}^{*}f_{j}$ and thus $Qf=\sum_{j\in
J}Q\mathcal{C}_{V,j}^{*}f_{j}$ by the boundedness of $Q$. Furthermore, since
$Qf\in\mathcal{K}_{W}^{2}$, we may write $Qf=([Qf]_{i})_{i\in I}$, where
$[Qf]_{i}\in W_{i}$ denotes the $i$-th component of $Qf$. By definition of the
coordinate functions $\mathcal{C}_{W,i}$ we obtain

$\displaystyle Qf=([Qf]_{i})_{i\in I}$
$\displaystyle=(\mathcal{C}_{W,i}Qf)_{i\in I}$
$\displaystyle=\Big{(}\mathcal{C}_{W,i}\sum_{j\in
J}Q\mathcal{C}_{V,j}^{*}f_{j}\Big{)}_{i\in I}=\Big{(}\sum_{j\in
J}\mathcal{C}_{W,i}Q\mathcal{C}_{V,j}^{*}f_{j}\Big{)}_{i\in I}.$

We set
$Q_{ij}:=\mathcal{C}_{W,i}Q\mathcal{C}_{V,j}^{*}\in\mathcal{B}(V_{j},W_{i})$
and observe that $Q$ acts on $f=(f_{j})_{j\in J}\in\mathcal{K}_{V}^{2}$ in the
same way as the matrix $[Q_{ij}]_{j\in J,i\in I}$ does in (19), when
considering $f$ as column vector. The uniqueness of the matrix $[Q_{ij}]_{j\in
J,i\in I}$ follows from its construction.

Note that we can also define a matrix representation of operators using fusion
frames as a generalization of the ideas in the above proofs, where not the
’canonical’ operators $\mathcal{C}_{W,i}$ and $\mathcal{C}_{V,j}$ are used,
but the fusion frame-related analysis and synthesis operators are, see
xxlcharshaare18 .

Next we show the converse of the above result. Its proof relies heavily on the
Uniform Boundedness Principle. In order to write down the proof properly, we
make use of the following projection operators defined below. Note that here
and in the subsequent proof, we may assume, without loss of generality, that
$I=J=\mathbb{N}$. For $m,n\in\mathbb{N}$ we define

$\displaystyle P_{\langle
n\rangle}^{V}:\mathcal{K}_{V}^{2}\longrightarrow\mathcal{K}_{V}^{2},\qquad$
$\displaystyle P_{\langle n\rangle}^{V}(f_{j})_{j\in
J}=(f_{1},...,f_{n},0,0,...),$ $\displaystyle P_{\langle
m\rangle}^{W}:\mathcal{K}_{W}^{2}\longrightarrow\mathcal{K}_{W}^{2},\qquad$
$\displaystyle P_{\langle m\rangle}^{W}(g_{i})_{i\in
I}=(g_{1},...,g_{m},0,0,...).$

It is easy to verify that $P_{\langle n\rangle}^{V}$ and $P_{\langle
m\rangle}^{W}$ are orthogonal projections ($m,n\in\mathbb{N}$).

###### Proposition 3

kohl21  Let $Q_{ij}\in\mathcal{B}(V_{j},W_{i})$ for all $j\in I,i\in I$, and
define $Q$ as in (19). If $\text{dom}(Q)=\mathcal{K}_{V}^{2}$, then
$Q\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$.

###### Proof

The assumption $\text{dom}(Q)=\mathcal{K}_{V}^{2}$ means that $Q$ is a well-
defined linear operator from $\mathcal{K}_{V}^{2}$ into $\mathcal{K}_{W}^{2}$.
Consider the sub-matrices $T_{m}=P_{\langle
m\rangle}^{W}Q:\mathcal{K}_{V}^{2}\longrightarrow\mathcal{K}_{W}^{2}$. If we
can show that each $T_{m}$ is bounded and that the operators $T_{m}$ converge
pointwise to $Q$ as $m\longrightarrow\infty$, then, by the Uniform Boundedness
Principle, this implies
$Q\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$.

To see that the operators $T_{m}$ converge pointwise to $Q$, recall that our
assumption $\text{dom}(Q)=\mathcal{K}_{V}^{2}$ implies that for all
$f\in\mathcal{K}_{V}^{2}$ we have $Qf\in\mathcal{K}_{W}^{2}$, which means that
$\|Qf\|_{\mathcal{K}_{W}^{2}}^{2}=\sum_{i=1}^{\infty}\|\sum_{j=1}^{\infty}Q_{ij}f_{j}\|_{W_{i}}^{2}<\infty$.
In particular, this implies

$\displaystyle\lim_{m\longrightarrow\infty}\|(Q-T_{m})f\|_{\mathcal{K}_{W}^{2}}^{2}$
$\displaystyle=\lim_{m\longrightarrow\infty}\|(\mathcal{I}_{\mathcal{K}_{W}^{2}}-P_{\langle
m\rangle}^{W})Qf\|_{\mathcal{K}_{W}^{2}}^{2}$
$\displaystyle=\lim_{m\longrightarrow\infty}\sum_{i=m+1}^{\infty}\Big{\|}\sum_{j=1}^{\infty}Q_{ij}f_{j}\Big{\|}_{W_{i}}^{2}=0.$

It remains to show that
$T_{m}\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$ for each $m$.
We will use the Uniform Boundedness Principle once again to see that this is
true. To this end, we fix an arbitrary $m\in\mathbb{N}$ and consider the sub-
matrices $T_{m}^{(n)}=P_{\langle m\rangle}^{W}QP_{\langle
n\rangle}^{V}:\mathcal{K}_{V}^{2}\longrightarrow\mathcal{K}_{W}^{2}$ of
$T_{m}$. To see that $T_{m}^{(n)}$ is bounded for each $n\in\mathbb{N}$,
observe that we have

$\displaystyle\|P_{\langle m\rangle}^{W}QP_{\langle
n\rangle}^{V}f\|_{\mathcal{K}_{W}^{2}}^{2}$
$\displaystyle=\sum_{i=1}^{K}\Big{\|}\sum_{j=1}^{n}Q_{ij}f_{j}\Big{\|}_{W_{i}}^{2}$
$\displaystyle\leq\sum_{i=1}^{K}\Big{(}\sum_{j=1}^{n}\|Q_{ij}\|_{V_{j}\rightarrow
W_{i}}\,\|f_{j}\|_{V_{j}}\Big{)}^{2}$
$\displaystyle\leq\sum_{i=1}^{K}\Big{(}\sum_{j=1}^{n}\|Q_{ij}\|_{V_{j}\rightarrow
W_{i}}^{2}\Big{)}\Big{(}\sum_{j=1}^{n}\|f_{j}\|_{V_{j}}^{2}\Big{)},$

where we used the Cauchy Schwartz inequality. This implies

$\|P_{\langle m\rangle}^{W}QP_{\langle
n\rangle}^{V}\|_{\mathcal{K}_{V}^{2}\rightarrow\mathcal{K}_{W}^{2}}^{2}\leq
mn\sup_{1\leq i\leq m,1\leq j\leq n}\|Q_{ij}\|_{V_{j}\rightarrow
W_{i}}^{2}<\infty.$

What remains to be proven is that $T_{m}^{(n)}\longrightarrow T_{m}$ pointwise
(as $n\longrightarrow\infty$), i.e. that for any $f\in\mathcal{K}_{V}^{2}$ we
have

$\lim_{n\longrightarrow\infty}\big{\|}(T_{m}-T_{m}^{(n)})f\big{\|}_{\mathcal{K}_{W}^{2}}=\lim_{n\longrightarrow\infty}\big{\|}P_{\langle
m\rangle}^{W}Q(\mathcal{I}_{\mathcal{K}_{V}^{2}}-P_{\langle
n\rangle}^{V})f\big{\|}_{\mathcal{K}_{W}^{2}}=0.$ (22)

To this end, fix an arbitrary $f=(f_{j})\in\mathcal{K}_{V}^{2}$ and define

$g^{(i)}_{n}:=\sum_{j=1}^{n}Q_{ij}f_{j}.$

Then clearly $g^{(i)}_{n}\in W_{i}$ for every $n\in\mathbb{N}$ and every $i$.
By construction we have
$\lim_{n\longrightarrow\infty}g^{(i)}_{n}=\sum_{j=1}^{\infty}Q_{ij}f_{j}\in
W_{i}$, which follows from $Qf\in\mathcal{K}_{W}^{2}$. Therefore, for every
$i$, the sequence $\\{x_{n}^{(i)}\\}_{n\in\mathbb{N}}$, defined by
$x_{n}^{(i)}=\|\sum_{j=n}^{\infty}Q_{ij}f_{j}\|_{W_{i}}^{2}$ is a convergent
sequence in $\mathbb{R}$ with limit $0$. This implies

$\displaystyle\lim_{n\longrightarrow\infty}\big{\|}P_{\langle
m\rangle}^{W}Q(\mathcal{I}_{\mathcal{K}_{V}^{2}}-P_{\langle
n\rangle}^{V})f\big{\|}_{\mathcal{K}_{W}^{2}}$
$\displaystyle=\lim_{n\rightarrow\infty}\sum_{i=1}^{K}\Big{\|}\sum_{j=n+1}^{\infty}Q_{ij}f_{j}\Big{\|}_{W_{i}}^{2}$
$\displaystyle=\sum_{i=1}^{K}\lim_{n\rightarrow\infty}\Big{\|}\sum_{j=n+1}^{\infty}Q_{ij}f_{j}\Big{\|}_{W_{i}}^{2}=0$

and (22) is proven.

We may summarize Propositions 2 and 3 as follows:

Every bounded operator
$Q\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$ can be naturally
identified with a matrix $[Q_{ij}]_{j\in J,i\in I}$ of bounded operators
$Q_{i}\in\mathcal{B}(V_{j},W_{i})$, which acts on $f=(f_{j})_{j\in J}$ via
matrix multiplication (as in (19)), and which additionally satisfies
$\text{dom}(Q)=\mathcal{K}_{V}^{2}$ (compare to (20)).

This motivates us to use the notation

$\mathbb{M}(Q)=[Q_{ij}]_{j\in J,i\in I}$

and call $\mathbb{M}(Q)=[Q_{ij}]_{j\in J,i\in I}$ the _canonical matrix
representation_ of $Q$.

###### Example 1

Examples Assume that $V=(V_{i},v_{i})_{i\in I}$ is a Bessel fusion sequence
for $\mathcal{H}$. Then, by Theorem 3.2,
$D_{V}\in\mathcal{B}(\mathcal{K}_{\mathcal{H}}^{2},\mathcal{H})$ and
$C_{V}\in\mathcal{B}(\mathcal{H},\mathcal{K}_{\mathcal{H}}^{2})$ are bounded
operators. Their respective canonical matrix representations are given by

$\displaystyle\mathbb{M}(D_{V})$
$\displaystyle=\begin{bmatrix}\dots&v_{i-1}\pi_{V_{i-1}}&v_{i}\pi_{V_{i}}&v_{i+1}\pi_{V_{i+1}}&\dots\end{bmatrix},$
$\displaystyle\mathbb{M}(C_{V})$
$\displaystyle=\begin{bmatrix}\dots&v_{i-1}\pi_{V_{i-1}}&v_{i}\pi_{V_{i}}&v_{i+1}\pi_{V_{i+1}}&\dots\end{bmatrix}^{T},$

where the exponent $T$ denotes matrix transposition, as usual.

Note: For the remainder of this subsection, we assume that
$\mathcal{K}_{V}^{2}$ and $\mathcal{K}_{W}^{2}$ are Hilbert direct sums
indexed by the _same_ index set $I$.

One particularly interesting subclass of operators between Hilbert direct sums
are the _block-diagonal_ operators (also known as _direct sums of operators_
conw1 ) between Hilbert direct sums associated to the same index set $I$. For
given operators $Q_{i}:\text{dom}(Q_{i})\subseteq V_{i}\longrightarrow W_{i}$,
the _block-diagonal_ operator $\bigoplus_{i\in I}Q_{i}$ is defined by

$\bigoplus_{i\in I}Q_{i}:\text{dom}\Big{(}\bigoplus_{i\in
I}Q_{i}\Big{)}\subseteq\mathcal{K}_{V}^{2}\longrightarrow\mathcal{K}_{W}^{2},\qquad\bigoplus_{i\in
I}Q_{i}(f_{i})_{i\in I}:=(Q_{i}f_{i})_{i\in I}.$ (23)

Here in this manuscript we only consider the case, where each
$Q_{i}\in\mathcal{B}(V_{i},W_{i})$ is bounded, see Section 6.

By their definition, bounded block-diagonal operators are precisely those
bounded operators in $\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$,
whose canonical matrix representation is given by a diagonal matrix, i.e.
$\mathbb{M}(\bigoplus_{i\in I}Q_{i})=\text{diag}(Q_{i})_{i\in I}$. The next
lemma gives an easy characterization for block-diagonal operators
$\bigoplus_{i\in I}Q_{i}$ associated to $Q_{i}\in\mathcal{B}(V_{i},W_{i})$
($i\in I$) being bounded.

###### Lemma 8

Assume that we are given bounded operators $Q_{i}\in\mathcal{B}(V_{i},W_{i})$
($i\in I$). Then $\bigoplus_{i\in I}Q_{i}$, defined as in (23), is a bounded
operator $\bigoplus_{i\in
I}Q_{i}\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$ if and only if
$\sup_{i\in I}\|Q_{i}\|<\infty$. In that case, $\|\bigoplus_{i\in
I}Q_{i}\|=\sup_{i\in I}\|Q_{i}\|<\infty$.

###### Proof

This proof merely consists of writing down the definitions and is therefore
left to the reader, see also kohl21  for more details.

###### Remark 3

A short computation shows that if $\bigoplus_{i\in
I}Q_{i}\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$ is bounded and
block-diagonal, then its adjoint $\big{(}\bigoplus_{i\in
I}Q_{i}\big{)}^{*}=\bigoplus_{i\in
I}Q_{i}^{*}\in\mathcal{B}(\mathcal{K}_{W}^{2},\mathcal{K}_{V}^{2})$ is again
block-diagonal koeba23 .

In this manuscript, we will also consider a sub-class of bounded block-
diagonal operators: We call a bounded block-diagonal operator $\bigoplus_{i\in
I}Q_{i}\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$ _component
preserving_ , if each $Q_{i}\in\mathcal{B}(V_{i},W_{i})$ is surjective.

By considering the self-adjoint operators
$M_{V,k}=\mathcal{C}_{V,k}^{*}\mathcal{C}_{V,k}\in\mathcal{B}(\mathcal{K}_{V}^{2})$,
given by $M_{V,k}(f_{i})_{i\in I}=(\delta_{ki}f_{i})_{i\in I}$ for $k\in I$,
and analogously
$M_{W,k}=\mathcal{C}_{W,k}^{*}\mathcal{C}_{W,k}\in\mathcal{B}(\mathcal{K}_{W}^{2})$,
we can state the following characterization of bounded operators
$Q\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$ being block-
diagonal (resp. component preserving). Its proof follows immediately from the
uniqueness of the canonical matrix representation of $Q$, see above.

###### Lemma 9

Let $Q\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{K}_{W}^{2})$. Then the
following are equivalent:

  1. (i)

$Q$ is block-diagonal.

  2. (ii)

$QM_{V,k}=M_{W,k}Q$ for each $k\in I$.

  3. (iii)

$QM_{V,k}(\mathcal{K}_{V}^{2})\subseteq M_{W,k}(\mathcal{K}_{W}^{2})$ for each
$k\in I$.

Furthermore, the following two conditions are equivalent:

  1. (iv)

$Q$ is component preserving.

  2. (v)

$QM_{V,k}(\mathcal{K}_{V}^{2})=M_{W,k}(\mathcal{K}_{W}^{2})$ for each $k\in
I$.

Note, that for the remainder of this survey, we reconsider the special cases
$\mathcal{K}_{V}^{2}=\mathcal{K}_{\mathcal{H}}^{2}$ and
$\mathcal{K}_{V}^{2}=\big{(}\sum_{i\in I}\oplus V_{i}\big{)}_{\ell^{2}}$,
where each $V_{i}$ is a closed subspace of $\mathcal{H}$, only. In particular,
in Section 6, we will not distinguish between $M_{V,k}$ and $M_{W,k}$, and
simply write $M_{k}$ instead.

##  4 Other notions in fusion frame theory

In this section we discuss further notions in fusion frame theory.

###### Definition 4

caskut04 ; Shaarbal  Let $(V_{i})_{i\in I}$ be a family of closed non-zero
subspaces of $\mathcal{H}$ and let $(v_{i})_{i\in I}$ be a family of weights.

  * •

$V=(V_{i},v_{i})_{i\in I}$ is called a _fusion Riesz basis_ , if
$\overline{\text{span}}(V_{i})_{i\in I}=\mathcal{H}$ and if there exist
constants $0<\alpha_{V}\leq\beta_{V}<\infty$, called _lower_ and _upper fusion
Riesz bounds_ respectively, such that for all finite vector-sequences
$(f_{i})_{i\in I}\in\mathcal{K}_{V}^{00}$ it holds

$\alpha_{V}\sum_{i\in I}\|f_{i}\|^{2}\leq\bigg{\|}\sum_{i\in
I}v_{i}f_{i}\bigg{\|}^{2}\leq\beta_{V}\sum_{i\in I}\|f_{i}\|^{2}.$ (24)

  * •

$(V_{i})_{i\in I}$ is called an _orthonormal fusion basis_ for $\mathcal{H}$,
if $\mathcal{H}=\bigoplus^{\perp}_{i\in I}V_{i}$ 111We use the notation
$\bigoplus^{\perp}_{i\in I}$ to emphasize that the subspaces are mutually
orthogonal, so we consider a very special direct sum., i.e.
$\mathcal{I}_{\mathcal{H}}=\sum_{i\in I}\pi_{V_{i}}$ and $V_{i}\perp V_{j}$
for all $i\neq j\in I$. In other words, an orthonormal fusion basis is a
family of pairwise orthogonal subspaces, so that
$S_{V^{1}}=\mathcal{I}_{\mathcal{H}}$, where $V^{1}=(V_{i},1)_{i\in I}$.
Therefore, we always consider uniform weights $v_{i}=1$ ($i\in I$) in this
case.

  * •

$(V_{i})_{i\in I}$ is called a _Riesz decomposition_ of $\mathcal{H}$, if for
every $f\in\mathcal{H}$ there exists a unique family $(f_{i})_{i\in I}$ of
vectors $f_{i}\in V_{i}$ ($i\in I$), such that $f=\sum_{i\in I}f_{i}$.

  * •

$(V_{i})_{i\in I}$ is called _minimal_ , if for each $i\in I$,
$V_{i}\cap\overline{\text{span}}(V_{j})_{j\in I,j\neq i}=\\{0\\}.$

At first, we show that every fusion Riesz basis is a fusion frame. This fact
follows from the following characterization of fusion Riesz bases via the
fusion frame-related operators $D_{V}$ and $C_{V}$.

###### Theorem 4.1

Assume that $V=(V_{i},v_{i})_{i\in I}$ is a weighted sequence of closed
subspaces $V_{i}$ of $\mathcal{H}$. Then the following are equivalent.

  1. (i)

$V$ is a fusion Riesz basis for $\mathcal{H}$.

  2. (ii)

$D_{V}$ is bounded on $\mathcal{K}_{\mathcal{H}}^{2}$ with
$\mathcal{R}(D_{V})=\mathcal{H}$ and
$\mathcal{N}(D_{V})=(\mathcal{K}_{V}^{2})^{\perp}$.

  3. (iii)

$C_{V}$ is bounded on $\mathcal{H}$ with
$\mathcal{R}(C_{V})=\mathcal{K}_{V}^{2}$ and $\mathcal{N}(C_{V})=\\{0\\}$.

In particular, every fusion Riesz basis is a fusion frame with the same
bounds.

###### Proof

First, note that by Lemma 6, the orthogonal complement
$(\mathcal{K}_{V}^{2})^{\perp}$ of $\mathcal{K}_{V}^{2}$ in
$\mathcal{K}_{\mathcal{H}}^{2}$ is given by $\big{(}\sum_{i\in I}\oplus
V_{i}^{\perp}\big{)}_{\ell^{2}}$. By definition of $D_{V}$, this implies that
$D_{V}=D_{V}(\pi_{\mathcal{K}_{V}^{2}}+\pi_{(\mathcal{K}_{V}^{2})^{\perp}})=D_{V}\pi_{\mathcal{K}_{V}^{2}}$,
and thus $D_{V}|_{\mathcal{K}_{V}^{2}}=D_{V}\pi_{\mathcal{K}_{V}^{2}}$ on
$\mathcal{K}_{V}^{2}$. In particular, statement (ii) is equivalent to
$D_{V}|_{\mathcal{K}_{V}^{2}}\in\mathcal{B}(\mathcal{K}_{V}^{2},\mathcal{H})$
being bijective and statement (iii) is equivalent to
$C_{V}\in\mathcal{B}(\mathcal{H},\mathcal{K}_{V}^{2})$ being bijective.
Moreover, as in the proof of Lemma 3, we can show that
$(D_{V}|_{\mathcal{K}_{V}^{2}})^{*}=C_{V}$ and we have
$C_{V}^{*}=D_{V}|_{\mathcal{K}_{V}^{2}}$ (assuming they are bounded), which
stems from the fact, that we always have
$\mathcal{R}(C_{V})\subseteq\mathcal{K}_{V}^{2}$.

(ii) $\Leftrightarrow$ (iii) This is generally true for a bounded bijective
operator between Hilbert spaces and its adjoint conw1 . As we will see in the
following all conditions imply that the operators are bounded.

(i) $\Rightarrow$ (ii) If $V$ is a fusion Riesz basis, then by the right-hand
inequality of (24), $D_{V}$ is bounded on $\mathcal{K}_{\mathcal{H}}^{00}$ and
thus by density also on $\mathcal{K}_{\mathcal{H}}^{2}$. This implies that
$C_{V}$ is also bounded. If $D_{V}|_{\mathcal{K}_{V}^{2}}$ was not surjective,
there would exist $0\neq
h\in\mathcal{R}(D_{V}|_{\mathcal{K}_{V}^{2}})^{\perp}=\mathcal{N}(C_{V})$ such
that $h\perp V_{i}$ for all $i\in I$. However, since $(V_{i})_{i\in I}$ is
complete by assumption, there exists some sequence
$(g_{n})_{n=1}^{\infty}\in\text{span}(V_{i})_{i\in I}$, such that

$0=\lim_{n\rightarrow\infty}\|h-g_{n}\|^{2}=\lim_{n\rightarrow\infty}(\|h\|^{2}-\langle
h,g_{n}\rangle-\langle g_{n},h\rangle+\|g_{n}\|^{2})=2\|h\|^{2}>0,$

a contradiction. Therefore $D_{V}|_{\mathcal{K}_{V}^{2}}$ is surjective. In
particular, $V$ is a fusion frame by Theorem 3.4 and thus series of the form
$D_{V}(f_{i})_{i\in I}=\sum_{i\in I}v_{i}f_{i}$ (with $(f_{i})_{i\in
I}\in\mathcal{K}_{V}^{2}$) converge unconditionally. To see that
$D_{V}|_{\mathcal{K}_{V}^{2}}$ is injective, observe that the left inequality
of (24) implies that $D_{V}|_{\mathcal{K}_{V}^{00}}$ is injective. However, by
continuity of $D_{V}$, density of $\mathcal{K}_{V}^{00}$ in
$\mathcal{K}_{V}^{2}$, and unconditional convergence of $\sum_{i\in
I}v_{i}f_{i}$, the left inequality of (24) is true for all sequences
$(f_{i})_{i\in I}\in\mathcal{K}_{V}^{2}$, which implies that
$D_{V}|_{\mathcal{K}_{V}^{2}}$ is injective.

$(ii)\Rightarrow(i)$ If
$D_{V}|_{\mathcal{K}_{V}^{2}}:\mathcal{K}_{V}^{2}\longrightarrow\mathcal{H}$
is bounded and bijective, then the inequalities (24) are true for all
sequences $(f_{i})_{i\in I}\in\mathcal{K}_{V}^{2}$ and hence for all finite
vector-sequences. If $(V_{i})_{i\in I}$ was not complete, then there would
exist $0\neq h\in\mathcal{H}$ such that
$h\perp\overline{\text{span}}(V_{i})_{i\in I}$. In particular, $C_{V}h=0$, a
contradiction. Thus $V$ is a fusion Riesz basis.

Finally, it follows from Theorem 3.4, that every fusion Riesz basis is a
fusion frame. In particular, if $V$ is a fusion Riesz basis (and thus a fusion
frame), then this means that $C_{V}$ is bounded from above and below by the
respective fusion frame bounds and $D_{V}$ is bounded from above and below by
the respective fusion Riesz basis bounds. Since $D_{V}C_{V}$ and $C_{V}D_{V}$
have the same non-zero spectrum, we can conclude that the set of fusion frame
bounds coincides with the set of fusion Riesz basis bounds.

Recall from Section 3, that the weights $(v_{i})_{i\in I}$ associated to a
fusion frame $V$ are uniformly bounded from above, but not necessarily semi-
normalized. However, weights associated to fusion Riesz bases are indeed semi-
normalized:

###### Lemma 10

Shaarbal  Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion Riesz basis with fusion
Riesz basis constants $\alpha_{V}$ and $\beta_{V}$. Then

  1. (i)

For all $i\in I$, $\sqrt{\alpha_{V}}\leq v_{i}\leq\sqrt{\beta_{V}}$

  2. (ii)

For every semi-normalized family $(w_{i})_{i\in I}$ of weights,
$(V_{i},v_{i}w_{i})_{i\in I}$ is a fusion Riesz basis as well.

###### Proof

(i) Recall that by the definition of a fusion Riesz basis, for any finite
vector sequence $(f_{j})_{j\in J}\in\mathcal{K}_{V}^{00}$, we have

$\alpha_{V}\sum_{j\in J}\|f_{j}\|^{2}\leq\Big{\|}\sum_{j\in
J}v_{j}f_{j}\Big{\|}^{2}\leq\beta_{V}\sum_{j\in J}\|f_{j}\|^{2}.$

Now, choose some arbitrary $i\in I$, set $J=\\{i\\}$, choose some normalized
$g\in V_{i}$ and apply the above.

(ii) follows analogously to the proof of Lemma 2.

Recall from frame theory, that every Parseval frame $(\varphi_{i})_{i\in I}$
with $\|\varphi_{i}\|=1$ ($\forall i\in I$) is an orthonormal basis (and vice
versa) ole1 . In the fusion frame setting, the analogue result is also true
(recall from Remark 1 (b), that for $1$-dimensional subspaces
$V_{i}=\text{span}\\{\varphi_{i}\\}$, $v_{i}$ corresponds to
$\|\varphi_{i}\|$).

###### Theorem 4.2

caskut04  Let $(V_{i})_{i\in I}$ be a family of closed subspaces of
$\mathcal{H}$. Then the following are equivalent.

  1. (i)

$(V_{i})_{i\in I}$ is an orthonormal fusion basis.

  2. (ii)

$(V_{i},1)_{i\in I}$ is a Parseval fusion frame.

###### Proof

(i)$\Rightarrow$(ii) If $(V_{i})_{i\in I}$ is an orthonormal fusion basis,
then for all $f\in\mathcal{H}$, $f=\sum_{i\in I}\pi_{V_{i}}f$ where the
subspaces $V_{i}$ are mutually orthogonal. In particular, this means, that the
fusion frame operator $S_{V^{1}}$ associated to $V^{1}=(V_{i},1)_{i\in I}$
satisfies $S_{V^{1}}=\mathcal{I}_{\mathcal{H}}$. By Corollary 4, this implies
that $V^{1}$ is a Parseval fusion frame.

(ii)$\Rightarrow$(i) For every $i\in I$, let $(e_{ij})_{j\in J_{i}}$ be an
orthonormal basis for $V_{i}$. If we assume that $(V_{i},1)_{i\in I}$ is a
Parseval fusion frame, then, by Corollary 2, the family $(e_{ij})_{i\in I,j\in
J_{i}}$ is a Parseval frame. Since every Parseval frame consisting of norm-$1$
vectors is an orthonormal basis ole1 , $(e_{ij})_{i\in I,j\in J_{i}}$ is an
orthonormal basis for $\mathcal{H}$. This implies that $V_{i}\perp V_{j}$ for
all $i\neq j$ and that $f=\sum_{i\in I}\pi_{V_{i}}f$ for all
$f\in\mathcal{H}$. Thus $(V_{i})_{i\in I}$ is an orthonormal fusion basis.

The next proposition is a slightly more general version of (caskut04, , Lemma
4.2), characterizing minimal families of subspaces using local Riesz bases
with a new proof.

###### Proposition 4

Let $(V_{i})_{i\in I}$ be a family of closed subspaces in $\mathcal{H}$ and
assume that $(\varphi_{ij})_{j\in J_{i}}$ is a Riesz basis for $V_{i}$ for
every $i\in I$. Then the following are equivalent.

  1. (i)

$(V_{i})_{i\in I}$ is minimal.

  2. (ii)

$(\varphi_{ij})_{i\in I,j\in J_{i}}$ is a minimal sequence in $\mathcal{H}$.

###### Proof

Note that for every subset $K\subset I$, $\overline{\text{span}}(V_{i})_{i\in
K}=\overline{\text{span}}(\varphi_{ij})_{i\in K,j\in J_{i}}$: Indeed, since
the families $(\varphi_{ij})_{j\in J_{i}}$ are local Riesz bases, we can
deduce that $\text{span}(V_{i})_{i\in
K}\subseteq\overline{\text{span}}(\varphi_{ij})_{i\in K,j\in J_{i}}$ and
$\text{span}(\varphi_{ij})_{i\in K,j\in
J_{i}}\subseteq\overline{\text{span}}(V_{i})_{i\in I}$. Now take the
respective closures.

(ii)$\Rightarrow$(i) If $(V_{i})_{i\in I}$ was not minimal, then there would
exist some non-zero $g\in\mathcal{H}$ with $g\in
V_{k}\cap\overline{\text{span}}(V_{i})_{i\in I,i\neq k}$ for some $k\in I$.
Because $(\varphi_{kj})_{j\in J_{k}}$ is a Riesz basis for $V_{k}$, we can
write $g=\sum_{j\in J_{k}}c_{kj}\varphi_{kj}$ for suitable scalars $c_{kj}$.
Since $g\neq 0$, there is a $c_{km}\neq 0$ and we have

$\displaystyle c_{km}\varphi_{km}$ $\displaystyle\in$
$\displaystyle\overline{\text{span}}(\varphi_{ij})_{i\neq k,j\in
J_{i}}-\sum_{j\neq m}c_{kj}\varphi_{kj}$ $\displaystyle\subseteq$
$\displaystyle\overline{\text{span}}(\varphi_{ij})_{i\neq k,j\in
J_{i}}+\overline{\text{span}}(\varphi_{kj})_{j\neq m}$
$\displaystyle\subseteq$
$\displaystyle\overline{\text{span}}(\varphi_{ij})_{i\in I,j\in
J_{i},(i,j)\neq(k,m)}.$

This contradicts (ii).

(i)$\Rightarrow$(ii) If $(\varphi_{ij})_{i\in I,j\in J_{i}}$ was not minimal,
then there would exist some non-zero
$\varphi_{kl}\in\overline{\text{span}}(\varphi_{ij})_{i\in I,j\in
J_{i},(i,j)\neq(k,l)}$. Since $(\varphi_{ij})_{j\in J_{i}}$ is a minimal
sequence in $V_{i}$ for every $i\in I$, we see that
$\varphi_{kl}\not\in\overline{\text{span}}(\varphi_{kj})_{j\in J_{k},j\neq
l}$, which implies $\varphi_{kl}\in\overline{\text{span}}(\varphi_{ij})_{i\in
I,j\in J_{i},i\neq k}=\overline{\text{span}}(V_{i})_{i\in I,i\neq k}$, a
contradiction.

Now, we are almost ready to characterize those fusion frames, that are fusion
Riesz bases. We only need to prove the following preparatory lemma.

###### Lemma 11

koeba23  Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted family of closed
subspaces of $\mathcal{H}$ and assume, that for each $i\in I$,
$\varphi^{(i)}=(\varphi_{ij})_{j\in J_{i}}$ is a Parseval frame for $V_{i}$.
Then $V$ is a fusion frame for $\mathcal{H}$ if and only if
$v\varphi=(v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$ is a frame for
$\mathcal{H}$. In this case, $S_{v\varphi}=S_{V}$.

###### Proof

The first part of the statement follows immediately from Theorem 3.1. For the
second part, recall that $\varphi^{(i)}$ is Parseval if and only if
$S_{\varphi^{(i)}}=\mathcal{I}_{V_{i}}$. Hence, for every $f\in\mathcal{H}$

$\displaystyle S_{v\varphi}f$ $\displaystyle=\sum_{i\in I}\sum_{j\in
J_{i}}\langle f,v_{i}\varphi_{ij}\rangle v_{i}\varphi_{ij}$
$\displaystyle=\sum_{i\in I}v_{i}^{2}\sum_{j\in J_{i}}\langle
f,\varphi_{ij}\rangle S_{\varphi^{(i)}}^{-1}\varphi_{ij}$
$\displaystyle=\sum_{i\in I}v_{i}^{2}\pi_{V_{i}}f=S_{V}f$

and the proof is complete.

After this preparation we can prove a slight generalization of a result in
Shaarbal :

###### Theorem 4.3

Let $(V_{i},v_{i},\varphi^{(i)})_{i\in I}$ be a fusion frame system, and let
$\varphi^{(i)}$ be a Riesz basis for $V_{i}$, for every $i\in I$. Then the
following are equivalent.

  1. (i)

$(V_{i})_{i\in I}$ is minimal in $\mathcal{H}$.

  2. (ii)

$(V_{i})_{i\in I}$ is a Riesz decomposition for $\mathcal{H}$.

  3. (iii)

$(V_{i},v_{i})_{i\in I}$ is a fusion Riesz basis for $\mathcal{H}$.

  4. (iv)

$(v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$ is a Riesz basis for $\mathcal{H}$.

  5. (v)

$S_{V}^{-1}V_{i}\perp V_{k}$, for all $i,k\in I$ with $i\neq k$.

  6. (vi)

$v_{i}^{2}\pi_{V_{k}}S_{V}^{-1}\pi_{V_{i}}=\delta_{ik}\pi_{V_{i}}$ for all
$i,k\in I$.

In this case, $(S_{V}^{-1}S_{\varphi^{(i)}}^{-1}v_{i}\varphi_{ij})_{i\in
I,j\in J_{i}}$ coincides with the canonical dual frame of
$(v_{i}\varphi_{ij})_{j\in J_{i}.i\in I}$.

###### Proof

(i)$\Rightarrow$(ii) Assume that $(V_{i})_{i\in I}$ is not a Riesz
decomposition of $\mathcal{H}$. Then there exists some $f\in\mathcal{H}$ with
$f=\sum_{i\in I}f_{i}=\sum_{i\in I}g_{i}$, where $(f_{i})_{i\in
I}\neq(g_{i})_{i\in I}\in(V_{i})_{i\in I}$. So, there exists some $k\in I$
with $f_{k}\neq g_{k}\in V_{k}$. This implies, that $0\neq
g_{k}-f_{k}=\sum_{i\in I,i\neq k}(f_{i}-g_{i})$ and hence $g_{k}-f_{k}\in
V_{k}\cap\overline{\text{span}}(V_{i})_{i\in I,i\neq k}$, which means that
$(V_{i})_{i\in I}$ is not minimal.

(ii)$\Rightarrow$(iii) By assumption, $V$ is a fusion frame, which, according
to Theorem 3.4, means that $D_{V}$ is bounded and surjective. Since
$\mathcal{R}(D_{V}|_{\mathcal{K}_{V}^{2}})=\mathcal{R}(D_{V})=\mathcal{H}$, we
particularly see that $D_{V}|_{\mathcal{K}_{V}^{2}}$ is bounded and
surjective. Now, if $(V_{i})_{i\in I}$ is a Riesz decomposition, then
$D_{V}|_{\mathcal{K}_{V}^{2}}$, by definition, is injective as well. By
Theorem 4.1, this implies that $V$ is a fusion Riesz basis.

(iii)$\Rightarrow$(i) If $V$ is a fusion Riesz basis, then, by Lemma 10,
$V^{1}:=(V_{i},1)_{i\in I}$ is a fusion Riesz basis as well. By Theorem 4.1,
its associated synthesis operator $D_{V^{1}}$ maps $\mathcal{K}_{V}^{2}$
boundedly and bijectively onto $\mathcal{H}$. Hence, for every
$f\in\mathcal{H}$, there exists a unique sequence $(f_{i})_{i\in
I}\in\mathcal{K}_{V}^{2}$, such that $f=D_{V^{1}}(f_{i})_{i\in I}=\sum_{i\in
I}f_{i}$. If was not $(V_{i})_{i\in I}$ is not minimal, then there there would
exist such an $f$ so that there exists another sequence $(g_{i})_{i\in
I}\in(V_{i})_{i\in I}$ satisfying $\sum_{i\in I}g_{i}=f$. Then, by definition
of $D_{V^{1}}$, $(g_{i})_{i\in
I}\in\text{dom}(D_{V^{1}})=\mathcal{K}_{V}^{2}$, which contradicts the
bijectivity of $D_{V^{1}}$.

(iii)$\Leftrightarrow$(iv) If (iii) is satisfied, then $(v_{i})_{i\in I}$ is
semi-normalized by Lemma 10. On the other hand, if (iv) holds, then
$(v_{i})_{i\in I}$ is semi-normalized as well: Indeed, similarly to Lemma 10,
one can show, that the frame vectors associated to a Riesz basis are semi-
normalized with uniform bounds given by the respective square-roots of the
frame bounds. Therefore (compare with Definition 3)
$\sqrt{A}\leq\sqrt{A_{i}}\leq\|\varphi_{ij}\|\leq\sqrt{B_{i}}\leq\sqrt{B}$
(for all $i\in I,j\in J_{i}$). At the same time we know that
$m\leq\|v_{i}\varphi_{ij}\|\leq M$ for suitable constants $m,M>0$ (for all
$i\in I,j\in J_{i}$). Thus $(v_{i})_{i\in I}$ must be semi-normalized. Now,
observe that (iv) holds if and only if $(v_{i}\varphi_{ij})_{i\in I,j\in
J_{i}}$ is a minimal sequence. Since $(v_{i})_{i\in I}$ is semi-normalized in
either case, the latter is true if and only if $(\varphi_{ij})_{i\in I,j\in
J_{i}}$ is minimal. By Proposition 4, this is true if and only if
$(V_{i})_{i\in I}$ is minimal, and by the equivalence (i) $\Leftrightarrow$
(iii) this is equivalent to $V$ being a fusion Riesz basis.

(iii)$\Rightarrow$(v) For every $i\in I$, let $(e_{ij})_{j\in J_{i}}$ be an
orthonormal basis for $V_{i}$. By the equivalence (iii)$\Leftrightarrow$(iv)
and Lemma 11, $ve:=(v_{i}e_{ij})_{i\in I,j\in J_{i}}$ is a Riesz basis for
$\mathcal{H}$ with $S_{ve}=S_{V}$. By Theorem 2.7, $ve$ has a unique
biorthogonal sequence given by $(S_{ve}^{-1}v_{i}e_{ij})_{i\in I,j\in
J_{i}}=(S_{V}^{-1}v_{i}e_{ij})_{i\in I,j\in J_{i}}$, which implies
$S_{V}^{-1}e_{ij}\perp e_{kl}$, whenever $(i,j)\neq(k,l)$. This yields
$S_{V}^{-1}V_{i}\perp V_{k}$ whenever $i\neq k$, as desired.

(v)$\Rightarrow$(vi) For each $k\in I$ we have

$\pi_{V_{k}}=\pi_{V_{k}}S_{V}^{-1}S_{V}=\sum_{i\in
I}v_{i}^{2}\pi_{V_{k}}S_{V}^{-1}\pi_{V_{i}}=v_{k}^{2}\pi_{V_{k}}S_{V}^{-1}\pi_{V_{k}}.$

(vi)$\Rightarrow$(v) Let $f\in V_{i}$ and $g\in V_{k}$, where $i\neq k$. Then

$\left<S_{V}^{-1}f,g\right>=\left<S_{V}^{-1}\pi_{V_{i}}f,\pi_{V_{k}}g\right>=\frac{1}{v_{i}^{2}}\left<v_{i}^{2}\pi_{V_{k}}S_{V}^{-1}\pi_{V_{i}}f,g\right>=0.$

(vi)$\Rightarrow$(i) By Theorem 2.7, it suffices to show that the frame
$(v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$ is biorthogonal to the family
$(S_{V}^{-1}S_{\varphi^{(i)}}^{-1}v_{i}\varphi_{ij})_{i\in I,j\in J_{i}}$. By
using condition (vi), we see that

$\displaystyle\langle
v_{i}\varphi_{ij},S_{V}^{-1}S_{\varphi^{(k)}}^{-1}v_{k}\varphi_{kl}\rangle$
$\displaystyle=\langle
v_{i}\pi_{V_{i}}\varphi_{ij},S_{V}^{-1}\pi_{V_{k}}S_{\varphi^{(k)}}^{-1}v_{k}\varphi_{kl}\rangle$
$\displaystyle=\frac{1}{v_{i}}\langle
v_{i}^{2}\pi_{V_{k}}S_{V}^{-1}\pi_{V_{i}}\varphi_{ij},S_{\varphi^{(k)}}^{-1}v_{k}\varphi_{kl}\rangle$
$\displaystyle=\frac{1}{v_{i}}\langle\delta_{ik}\pi_{V_{i}}\varphi_{ij},S_{\varphi^{(k)}}^{-1}v_{k}\varphi_{kl}\rangle=\frac{1}{v_{i}}\langle\delta_{ik}\varphi_{ij},S_{\varphi^{(k)}}^{-1}v_{k}\varphi_{kl}\rangle.$

In case $k\neq i$ this gives $\langle
v_{i}\varphi_{ij},S_{V}^{-1}S_{\varphi^{(k)}}^{-1}v_{k}\varphi_{kl}\rangle=0$
and in case $k=i$ we continue with $\langle
v_{i}\varphi_{ij},S_{V}^{-1}S_{\varphi^{(k)}}^{-1}v_{k}\varphi_{kl}\rangle=\frac{1}{v_{k}}\langle\varphi_{kj},S_{\varphi^{(k)}}^{-1}v_{k}\varphi_{kl}\rangle=\langle\varphi_{kj},S_{\varphi^{(k)}}^{-1}\varphi_{kl}\rangle=\delta_{jl}$,
where we used that each local Riesz basis $\varphi^{(i)}$ is biorthogonal to
its canonical dual $(S_{\varphi^{(i)}}^{-1}\varphi_{ij})_{j\in J_{i}}$.

The final statement follows from the proof of (vi)$\Rightarrow$(i) combined
with Theorem 2.7.

As a consequence we obtain the following.

###### Corollary 5

If $V=(V_{i},v_{i})_{i\in I}$ is a fusion Riesz basis such that $S_{V}$ and
$\pi_{V_{i}}$ commute for all $i\in I$, then $(V_{i})_{i\in I}$ is an
orthonormal fusion basis. In particular, every tight fusion Riesz basis is an
orthonormal fusion basis.

###### Proof

Using condition (vi) from Theorem 4.3, we obtain

$\sum_{i\in I}\pi_{V_{i}}=\sum_{i\in
I}v_{i}^{2}\pi_{V_{i}}S_{V}^{-1}\pi_{V_{i}}=S_{V}^{-1}S_{V}=\mathcal{I}_{\mathcal{H}},$

which means that $(V_{i},1)_{i\in I}$ is a Parseval fusion frame. By Theorem
4.2, this implies that $(V_{i})_{i\in I}$ is an orthonormal fusion basis.

The following result shows that implication (iii)$\Rightarrow$(i) of Theorem
2.7 does not hold in the fusion frame setting. Only the converse statement
remains true for fusion frames.

###### Proposition 5

caskut04  Every fusion Riesz basis is an exact fusion frame. The converse is
not true in general.

###### Proof

Let $V=(V_{i},v_{i})$ be a fusion Riesz basis. Then, by Theorem 4.1, $C_{V}$
maps $\mathcal{H}$ bijectively onto $\mathcal{K}_{V}^{2}$. Fix some arbitrary
index $k\in I$ and choose some non-zero $g\in V_{k}$. Then
$(\dots,0,0,g,0,0,\dots)$ ($g$ in the $k$-th component) is an element in
$\mathcal{K}_{V}^{2}$ and thus contained in $\mathcal{R}(C_{V})$. Therefore,
there is a unique non-zero $f\in\mathcal{H}$ such that
$(\dots,0,0,g,0,0,\dots)=C_{V}f=(v_{i}\pi_{V_{i}}f)_{i\in I}$. In particular
$\pi_{V_{i}}f=0$ for all $i\neq k$. Hence, if we removed $(V_{k},v_{k})$ from
the fusion frame $V$, we would not be able to maintain the fusion frame
inequalities for $f$. Since $k$ was chosen arbitrary, this means that $V$ is
exact.

To show that the converse statement is false, consider an orthonormal basis
$(e_{i})_{i=1}^{3}$ for $\mathbb{R}^{3}$, and let
$E_{1}=\text{span}\\{e_{1},e_{2}\\}$, $E_{2}=\text{span}\\{e_{2},e_{3}\\}$ and
$w_{1}=w_{2}=1$. It is immediate that this is an exact fusion frame and since
$E_{1}\cap E_{2}\neq 0$, it is not a Riesz decomposition and hence (Theorem
4.3) not a fusion Riesz basis. Alternatively, note that the global frame
$(e_{1},e_{2},e_{2},e_{3})$ is not a Riesz basis.

##  5 Fusion frames and operators

In this section, we consider subspaces under the action of operators and their
spanning properties. Here we will encounter more results, which show the
contrast between frames and fusion frames, such as Proposition 5.

We start with the following preliminary results, which directly popped up at
the first discussion of the duality of fusion frames and the related
observation that $T\pi_{V}$ is in general not a projection.

###### Lemma 12

cakuli08  Let $V$ be a closed subspace of $\mathcal{H}$ and
$T\in\mathcal{B}(\mathcal{H})$. Then

$\displaystyle\pi_{V}T^{*}=\pi_{V}T^{*}\pi_{\overline{TV}}.$ (25)

###### Proof

First note that $g\in(TV)^{\perp}=(\overline{TV})^{\perp}$ if and only if
$T^{*}g\in V^{\perp}$. Therefore, for all $f\in\mathcal{H}$, we see that

$\pi_{V}T^{*}f=\pi_{V}T^{*}\pi_{\overline{TV}}f+\pi_{V}T^{*}\pi_{(TV)^{\perp}}f=\pi_{V}T^{*}\pi_{\overline{TV}}f.$

###### Lemma 13

Let $U\in\mathcal{B}(\mathcal{H})$ and $(\varphi_{i})_{i\in
I}\subseteq\mathcal{H}$ be a countable family of vectors in $\mathcal{H}$.
Then $U\overline{\text{span}}(\varphi_{i})_{i\in
I}\subseteq\overline{\text{span}}(U\varphi_{i})_{i\in I}$. If $U$ is
additionally bijective, then $U\overline{\text{span}}(\varphi_{i})_{i\in
I}=\overline{\text{span}}(U\varphi_{i})_{i\in I}$.

###### Proof

Let $g=Uf\in U\overline{\text{span}}(\varphi_{i})_{i\in I}$ for
$f\in\overline{\text{span}}(\varphi_{i})_{i\in I}$. Then there exists a
sequence $(f_{n})_{n=1}^{\infty}\subseteq\text{span}(\varphi_{i})_{i\in I}$
such that $\|f-f_{n}\|\longrightarrow 0$ as $n\rightarrow\infty$. This implies
that $(Uf_{n})_{n=1}^{\infty}\in\text{span}(U\varphi_{i})_{i\in I}$ converges
to $Uf=g$ in $\mathcal{H}$, since $\|Uf-
Uf_{n}\|\leq\|U\|\|f-f_{n}\|\longrightarrow 0$ as $n\rightarrow\infty$. Thus
$g\in\overline{\text{span}}(U\varphi_{i})_{i\in I}$.

For the second part, assume that
$g\in\overline{\text{span}}(U\varphi_{i})_{i\in I}$. Then there exists a
sequence
$(g_{n})_{n=1}^{\infty}=(Uf_{n})_{n=1}^{\infty}\subseteq\text{span}(U\varphi_{i})_{i\in
I}$, such that $\|g-Uf_{n}\|\longrightarrow 0$ as $n\rightarrow\infty$. In
particular, we have
$(f_{n})_{n=1}^{\infty}\subseteq\text{span}(\varphi_{i})_{i\in I}$. Moreover,
since $U$ is surjective, there exists some $f\in\mathcal{H}$ with $Uf=g$. By
using the invertibility of $U$, this implies that
$f\in\overline{\text{span}}(\varphi_{i})_{i\in I}$, because
$\|f-f_{n}\|=\|U^{-1}Uf-U^{-1}Uf_{n}\|\leq\|U^{-1}\|\|Uf-
Uf_{n}\|\longrightarrow 0$ as $n\rightarrow\infty$. Thus $g=Uf\in
U\overline{\text{span}}(\varphi_{i})_{i\in I}$.

In case $U\in\mathcal{B}(\mathcal{H})$ is not bijective, then the reverse
inclusion in Lemma 13 is not true in general:

###### Example 2

Example Let $\mathcal{H}=\ell^{2}(\mathbb{N})$ and
$\varphi_{i}=(\delta_{in}\cdot
1/\sqrt{n})_{n\in\mathbb{N}}\in\ell^{2}(\mathbb{N})$ for $i\in\mathbb{N}$. The
operator $U:\ell^{2}(\mathbb{N})\longrightarrow\ell^{2}(\mathbb{N})$, defined
by $U(a_{n})_{n\in\mathbb{N}}=(a_{n}\cdot 1/\sqrt{n})_{n\in\mathbb{N}}$, is
bounded on $\ell^{2}(\mathbb{N})$ with $\|U\|=1$ and we have
$U\varphi_{i}=(\delta_{in}\cdot 1/n)_{n\in\mathbb{N}}$. Observe that
$(1/n)_{n\in\mathbb{N}}=\sum_{i\in\mathbb{N}}U\varphi_{i}\in\overline{\text{span}}(U\varphi_{i})_{i\in\mathbb{N}}$.
However, $(1/n)_{n\in\mathbb{N}}$ is not contained in
$U\overline{\text{span}}(\varphi_{i})_{i\in\mathbb{N}}$. Indeed, by component-
wise definition of $U$, the only possible sequence, which could be mapped onto
$(1/n)_{n\in\mathbb{N}}$ by $U$, is $(1/\sqrt{n})_{n\in\mathbb{N}}$. But since
$(1/\sqrt{n})_{n\in\mathbb{N}}\notin\ell^{2}(\mathbb{N})$, we particularly
have that
$(1/\sqrt{n})_{n\in\mathbb{N}}\notin\overline{\text{span}}(\varphi_{i})_{i\in\mathbb{N}}$.
Note that the above demonstrates, that $U$ is not surjective.

From Lemma 13, we can also easily deduce the following two results.

###### Corollary 6

Let $(\varphi_{i})_{i\in I}\subseteq\mathcal{H}$ and
$U\in\mathcal{B}(\mathcal{H})$.

  1. (i)

If $(U\varphi_{i})_{i\in I}$ is minimal and $U$ is injective, then
$(\varphi_{i})_{i\in I}$ is minimal. In case $U$ is bijective, the converse is
true as well.

  2. (ii)

If $(\varphi_{i})_{i\in I}$ is a complete sequence and $U$ has dense range,
then $(U\varphi_{i})_{i\in I}$ is a complete sequence. In case $U$ is
bijective, the converse is also valid.

###### Proof

(i) Assume $(U\varphi_{i})_{i\in I}$ is a minimal sequence and that
$(\varphi_{i})_{i\in I}$ is not minimal. Then there is some $m\in I$ such that
$\varphi_{m}\in\overline{\text{span}}(\varphi_{i})_{i\in I,i\neq m}$. Since
$U$ is injective, we have $U\varphi_{m}\neq 0$ and by Lemma 13,
$U\varphi_{m}\in U\overline{\text{span}}(\varphi_{i})_{i\in I,i\neq
m}\subseteq\overline{\text{span}}(U\varphi_{i})_{i\in I,i\neq m}$, a
contradiction. The second statement is shown similarly.

(ii) Since $(\varphi_{i})_{i\in I}$ is complete, we see by using Lemma 13 that
$U(\mathcal{H})=U\overline{\text{span}}(\varphi_{i})_{i\in
I}\subseteq\overline{\text{span}}(U\varphi_{i})_{i\in I}$. Thus
$\mathcal{H}=\overline{U(\mathcal{H})}\subseteq\overline{\text{span}}(U\varphi_{i})_{i\in
I}$. Analogously, the second statement is shown by repeating the above
argument with $U^{-1}$ instead of $U$.

###### Corollary 7

Let $V$ be a closed subspace of $\mathcal{H}$ and let
$U\in\mathcal{B}(\mathcal{H})$ be bijective. Then $UV$ is a closed subspace of
$\mathcal{H}$.

After these preparatory results, we can show that the fusion frame property is
invariant under the action of a bounded invertible operator. More precisely:

###### Theorem 5.1

cakuli08  Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion frame for $\mathcal{H}$
with fusion frame bounds $A_{V}\leq B_{V}$, and let
$U\in\mathcal{B}(\mathcal{H})$ be invertible. Then $UV=(UV_{i},v_{i})_{i\in
I}$ is a fusion frame for $\mathcal{H}$ with fusion frame bounds
$A_{V}\left\|U^{-1}\right\|^{-2}\|U\|^{-2}$ and
$B_{V}\left\|U^{-1}\right\|^{2}\left\|U\right\|^{2}$, and its associated
fusion frame operator $S_{UV}$ satisfies

$\frac{US_{V}U^{*}}{\|U\|^{2}}\leq S_{UV}\leq\|U^{-1}\|^{2}US_{V}U^{*}.$

###### Proof

By Corollary 7, each subspace $UV_{i}$ is closed. Let $f\in\mathcal{H}$ be
arbitrary. Then by (25), we have

$\displaystyle\langle S_{UV}f,f\rangle$ $\displaystyle=\sum_{i\in
I}v_{i}^{2}\left\|\pi_{UV_{i}}f\right\|^{2}$ $\displaystyle=\sum_{i\in
I}v_{i}^{2}\left\|\pi_{UV_{i}}(U^{-1})^{*}\pi_{V_{i}}U^{*}f\right\|^{2}$
$\displaystyle\leq\left\|U^{-1}\right\|^{2}\sum_{i\in
I}v_{i}^{2}\left\|\pi_{V_{i}}U^{*}f\right\|^{2}$
$\displaystyle=\left\|U^{-1}\right\|^{2}\Big{\langle}US_{V}U^{*}f,f\Big{\rangle}.$

By Lemma 12, we see that

$\displaystyle\Big{\langle}\frac{US_{V}U^{*}}{\|U\|^{2}}f,f\Big{\rangle}$
$\displaystyle=\frac{1}{\|U\|^{2}}\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}U^{*}f\|^{2}$
$\displaystyle=\frac{1}{\|U\|^{2}}\sum_{i\in
I}v_{i}^{2}\|\pi_{V_{i}}U^{*}\pi_{UV_{i}}f\|^{2}$ $\displaystyle\leq\sum_{i\in
I}v_{i}^{2}\|\pi_{UV_{i}}f\|^{2}=\langle S_{UV}f,f\rangle.$

Thus

$\frac{US_{V}U^{*}}{\|U\|^{2}}\leq S_{UV}\leq\|U^{-1}\|^{2}US_{V}U^{*}.$

Moreover, observe that $\|f\|\leq\|U^{-1}\|\|U^{*}f\|$ for all
$f\in\mathcal{H}$. Piecing all observations together, we obtain

$\displaystyle A_{V}\frac{1}{\|U^{-1}\|^{2}\|U\|^{2}}\|f\|^{2}$
$\displaystyle\leq A_{V}\frac{1}{\|U\|^{2}}\|U^{*}f\|^{2}$
$\displaystyle\leq\frac{1}{\|U\|^{2}}\Big{\langle}S_{V}U^{*}f,U^{*}f\Big{\rangle}$
$\displaystyle\leq\langle S_{UV}f,f\rangle$
$\displaystyle\leq\left\|U^{-1}\right\|^{2}\Big{\langle}US_{V}U^{*}f,f\Big{\rangle}\leq
B_{V}\left\|U^{-1}\right\|^{2}\|U\|^{2}\|f\|^{2}$

for all $f\in\mathcal{H}$, as desired.

From the above, we immediately obtain the following results:

###### Corollary 8

Gavruta7  Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted family of closed
subspaces in $\mathcal{H}$ and let $U\in\mathcal{B}(\mathcal{H})$ be
invertible. Then the following are equivalent:

  1. (i)

$V$ is a fusion frame for $\mathcal{H}$.

  2. (ii)

$UV=(UV_{i},v_{i})_{i\in I}$ is a fusion frame for $\mathcal{H}$.

###### Corollary 9

Gavruta7  Let $(V_{i},v_{i})_{i\in I}$ be a fusion frame for $\mathcal{H}$
with fusion frame bounds $A_{V}\leq B_{V}$. Then
$S_{V}^{-1}V=(S_{V}^{-1}V_{i},v_{i})_{i\in I}$ is a fusion frame for
$\mathcal{H}$ with fusion frame bounds $A_{V}^{3}/B_{V}^{2}\leq
B_{V}^{3}/A_{V}^{2}$.

###### Proof

Substituting $S_{V}^{-1}$ for $U$ in Theorem 5.1 yields that
$(S_{V}^{-1}V_{i},v_{i})_{i\in I}$ is a fusion frame with bounds
$A_{V}\|S_{V}\|^{-2}\|S_{V}^{-1}\|^{-2}$ and
$B_{V}\|S_{V}\|^{2}\|S_{V}^{-1}\|^{2}$. Combining this with the respective
operator norm estimates from (10) gives the claimed fusion frame bounds.

For unitary operators $U\in\mathcal{B}(\mathcal{H})$, Theorem 5.1 implies the
following.

###### Corollary 10

Let $V$ be a fusion frame for $\mathcal{H}$ and $U\in\mathcal{B}(\mathcal{H})$
be unitary. Then $UV$ is a fusion frame with the same fusion frame bounds and
its associated fusion frame operator is given by $S_{UV}=US_{V}U^{*}$.

We also note that fusion Riesz bases are invariant under the action of a
bounded invertible operator. In other words, Corollary 1 (iii) remains valid
in the fusion frame setting.

###### Theorem 5.2

Let $V=(V_{i},v_{i})_{i\in I}$ be a weighted family of closed subspaces in
$\mathcal{H}$ and let $U\in\mathcal{B}(\mathcal{H})$ be invertible. The
following are equivalent.

  1. (i)

$V$ is a fusion Riesz basis for $\mathcal{H}$.

  2. (ii)

$UV:=(UV_{i},v_{i})_{i\in I}$ is a fusion Riesz basis for $\mathcal{H}$.

###### Proof

(i)$\Rightarrow$(ii) Theorem 5.1 implies that $UV$ is a fusion frame. Hence,
according to Theorem 4.3, it suffices to show that $(UV_{i})_{i\in I}$ is
minimal. To this end, assume that $(\varphi_{ij})_{j\in J_{i}}$ be a Riesz
basis for $V_{i}$ for every $i\in I$. Since $V$ is a fusion Riesz basis,
Theorem 4.3 implies that $V$ is a minimal fusion frame. By Proposition 4,
$(\varphi_{ij})_{i\in I,j\in J_{i}}$ is a minimal sequence in $\mathcal{H}$,
and by Corollary 6, $(U\varphi_{ij})_{i\in I,j\in J_{i}}$ is minimal as well.
Since $U$ is invertible, it is also easy to see that $(U\varphi_{ij})_{j\in
J_{i}}$ is a Riesz basis for $UV_{i}$ for every $i\in I$. Thus $UV$ is minimal
by Proposition 4.

(i)$\Rightarrow$(ii) If $UV$ is a fusion Riesz basis, then so is $V=U^{-1}UV$
by the above.

In view of Corollary 1 (ii), it seems only natural to investigate fusion
frames under the action of a bounded surjective operator next. At first glance
one would guess, that if $V=(V_{i},v_{i})_{i\in I}$ is a fusion frame and
$U\in\mathcal{B}(\mathcal{H})$ surjective, then $UV=(UV_{i},v_{i})_{i\in I}$
is a fusion frame for $\mathcal{H}$, since the corresponding result for frames
is true as well. However, such a result cannot be achieved with fusion frames
in full generality. Indeed, for a surjective operator
$U\in\mathcal{B}(\mathcal{H})$, $UV_{i}$ is not even a closed subspace in
general. Even if we want to prove the corresponding result for the closed
subspaces $\overline{UV_{i}}$, we need to make an additional assumption,
because otherwise the statement is not valid (see below).

###### Proposition 6

Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion frame for $\mathcal{H}$ with fusion
frame bounds $A_{V}\leq B_{V}$, and let $U\in\mathcal{B}(\mathcal{H})$ be
surjective with the additional property that $U^{*}UV_{i}\subseteq V_{i}$ for
all $i\in I$. Then $(\overline{UV_{i}},v_{i})_{i\in I}$ is a fusion frame with
fusion frame bounds $A_{V}\left\|U^{{\dagger}}\right\|^{-2}\|U\|^{-2}\leq
B_{V}\left\|U^{{\dagger}}\right\|^{2}\|U\|^{2}$.

###### Proof

Let $(e_{ij})_{j\in J_{i}}$ be an orthonormal basis for $V_{i}$ for every
$i\in I$. Then, by Theorem 3.1, $(v_{i}e_{ij})_{i\in I,j\in J_{i}}$ is a frame
for $\mathcal{H}$ with frame bounds $A_{V}$ and $B_{V}$. By Theorem 2.8,
$(v_{i}Ue_{ij})_{i\in I,j\in J_{i}}$ is a frame for $\mathcal{H}$ as well and
one can compute (ch08, , Corollary 5.3.2) that
$A_{V}\left\|U^{{\dagger}}\right\|^{-2}$ and $B_{V}\|U\|^{2}$ are frame bounds
for it. Now, let $i\in I$ be arbitrary and choose $Uf_{i}\in UV_{i}$. Then

$\frac{\left\|Uf_{i}\right\|^{2}}{\left\|U^{{\dagger}}\right\|^{2}}=\frac{\left\|\pi_{\mathcal{R}(U)}^{*}Uf_{i}\right\|^{2}}{\left\|U^{{\dagger}}\right\|^{2}}=\frac{\left\|(U^{{\dagger}})^{*}U^{*}Uf_{i}\right\|^{2}}{\left\|U^{{\dagger}}\right\|^{2}}\leq\left\|U^{*}Uf_{i}\right\|^{2}.$

Since $U^{*}UV_{i}\subseteq V_{i}$, we may continue with

$\left\|U^{*}Uf_{i}\right\|^{2}=\sum_{j\in
J_{i}}\left|\left<U^{*}Uf_{i},e_{ij}\right>\right|^{2}=\sum_{j\in
J_{i}}\left|\left<Uf_{i},Ue_{ij}\right>\right|^{2}.$

On the other hand, clearly
$\left\|U^{*}Uf_{i}\right\|^{2}\leq\left\|U\right\|^{2}\left\|Uf_{i}\right\|^{2}$.
All together, this means that $Ue^{(i)}:=(Ue_{ij})_{j\in J_{i}}$ satisfies the
frame inequalities for all elements from $UV_{i}$. Finally, by using a density
argument (ole1, , Lemma 5.1.7), we can conclude that the families $Ue^{(i)}$
are frames for their respective closed subspaces $\overline{UV_{i}}$ with
common frame bounds $\|U^{\dagger}\|^{-2}$ and $\|U\|^{2}$. Hence, employing
Theorem 3.1, yields that $(\overline{UV_{i}},v_{i})$ is a fusion frame for
$\mathcal{H}$ with fusion frame bounds
$A_{V}\left\|U^{{\dagger}}\right\|^{-2}\|U\|^{-2}$ and
$B_{V}\left\|U^{{\dagger}}\right\|^{2}\|U\|^{2}$.

The following example demonstrates, that even if we consider an orthonormal
fusion basis $(E_{i})_{i\in I}$ and apply a bounded surjective operator
$U\in\mathcal{B}(\mathcal{H})$ on each $E_{i}$, then, different to the Hilbert
frame setting - Theorem 2.8 and Corollary 1 \- we don’t necessarily obtain
that $(\overline{UE_{i}},v_{i})_{i\in I}$ is a fusion frame, no matter which
family of weights $(v_{i})_{i\in I}$ we choose. This is another observation
that separates the properties of frames from the properties of fusion frames.

###### Example 3

Example rust08  Let $(e_{n})_{n\in\mathbb{N}}$ be an orthonormal basis for
some infinite-dimensional Hilbert space $\mathcal{H}$ and define
$E_{k}:=\text{span}\\{e_{2k-1},e_{2k}\\}$ for every $k\in\mathbb{N}$. Then
$(E_{k})_{k\in\mathbb{N}}$ is an orthonormal fusion basis.

Next, consider the linear operator
$U:\mathcal{H}_{0}\longrightarrow\mathcal{H}$ defined by

$Ue_{n}:=\begin{cases}2^{-\frac{n+1}{2}}e_{1}&\text{if }n\text{ is odd}\\\
e_{\frac{n}{2}+1}&\text{if }n\text{ is even}\end{cases},$

where, $\mathcal{H}_{0}:=\big{\\{}f\in\mathcal{H}:f=\sum_{n\in K}\langle
f,e_{n}\rangle e_{n},|K|<\infty\big{\\}}$. We show that $U$ is bounded on
$\mathcal{H}_{0}$. For arbitrary $g=\sum_{n\in K}\langle g,e_{n}\rangle
e_{n}\in\mathcal{H}_{0}$ we have

$\displaystyle Ug$ $\displaystyle=\sum_{n\in K}\langle g,e_{n}\rangle Ue_{n}$
$\displaystyle=\sum_{\begin{subarray}{c}n\in K\\\ n\text{
odd}\end{subarray}}\langle g,e_{n}\rangle
2^{-\frac{n+1}{2}}e_{1}+\sum_{\begin{subarray}{c}n\in K\\\ n\text{
even}\end{subarray}}\langle g,e_{n}\rangle e_{\frac{n}{2}+1}$

and thus

$\displaystyle\big{\|}Ug\big{\|}^{2}$
$\displaystyle=\left\|\sum_{\begin{subarray}{c}n\in K\\\ n\text{
odd}\end{subarray}}\langle g,e_{n}\rangle
2^{-\frac{n+1}{2}}e_{1}\right\|^{2}+\sum_{\begin{subarray}{c}n\in K\\\ n\text{
even}\end{subarray}}\big{\|}\langle g,e_{n}\rangle
e_{\frac{n}{2}+1}\big{\|}^{2}$
$\displaystyle\leq\left(\sum_{\begin{subarray}{c}n\in K\\\ n\text{
odd}\end{subarray}}|\langle
g,e_{n}\rangle|2^{-\frac{n+1}{2}}\right)^{2}+\sum_{\begin{subarray}{c}n\in
K\\\ n\text{ even}\end{subarray}}|\langle g,e_{n}\rangle|^{2}$
$\displaystyle\leq\left(\sum_{n\in K}|\langle
g,e_{n}\rangle|^{2}\right)\left(\sum_{n\in K}2^{-(n+1)}\right)+\sum_{n\in
K}|\langle g,e_{n}\rangle|^{2}$ $\displaystyle\leq 3\|g\|^{2}.$

Since $\mathcal{H}_{0}$ is a dense subspace of $\mathcal{H}$, there exists a
unique extension of $U$ (again denoted by $U$) to a bounded operator on
$\mathcal{H}$. To see that $U$ is surjective, let $f\in\mathcal{H}$ be
arbitrary. Then, by definition of $U$,

$\displaystyle f$ $\displaystyle=\sum_{n=1}^{\infty}\langle f,e_{n}\rangle
e_{n}$ $\displaystyle=\langle f,2Ue_{1}\rangle
2Ue_{1}+\sum_{n=2}^{\infty}\langle f,Ue_{2n-2}\rangle Ue_{2n-2}$
$\displaystyle=U\Big{(}4\langle U^{*}f,e_{1}\rangle
e_{1}+\sum_{n=2}^{\infty}\langle U^{*}f,e_{2n-2}\rangle e_{2n-2}\Big{)}.$

Now, set $V_{k}:=UE_{k}=\text{span}\\{e_{1},e_{k+1}\\}$ for $k\in\mathbb{N}$.
Towards a contradiction, suppose that there exists a family
$(v_{k})_{k\in\mathbb{N}}$ of weights, such that
$(V_{k},v_{k})_{k\in\mathbb{N}}$ is a fusion frame for $\mathcal{H}$ with
fusion frame bounds $A_{v}\leq B_{v}$. Then, by applying the corresponding
fusion frame inequalities to $f=e_{1}$, we obtain
$A_{v}\leq\sum_{k\in\mathbb{N}}v_{k}^{2}\leq B_{v}$, i.e
$(v_{k})_{k\in\mathbb{N}}\in\ell^{2}(\mathbb{N})$. On the other hand, for
$f=e_{k+1}$, we obtain

$A_{v}=A_{v}\|e_{k+1}\|^{2}\leq\sum_{l\in\mathbb{N}}v_{l}^{2}\|\pi_{V_{l}}e_{k+1}\|^{2}=v_{k}^{2}\qquad(\forall
k\in\mathbb{N}),$

which violates $v_{k}\rightarrow 0$ (as $k\rightarrow\infty$), a
contradiction.

The reason for the failure in the above example, is that
$\frac{\gamma(U\pi_{E_{k}})}{\|U\pi_{E_{k}}\|}=2^{-k}\rightarrow 0$ (as
$k\rightarrow\infty$), where $\gamma(A)$ denotes the _minimum modulus_ of a
bounded operator $A\in\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})$, defined
by $\gamma(A)=\inf\big{\\{}\|Af\|,\|f\|=1,f\in\mathcal{N}(A)^{\perp}\big{\\}}$
rust08 . Operator theoretic notions such as the minimum modulus can be used in
order to derive conditions, such that statements in the flavour of Proposition
6 (and analogous ones for fusion frame sequences) can be proven to be true
rust08 .

The converse of the above example is true in the following sense. Below, we
show that every fusion frame $V=(V_{i},v_{i})_{i\in I}$ contains the image of
an orthonormal fusion basis $(E_{i})_{i\in I}$ under a bounded surjective
operator $U\in\mathcal{B}(\mathcal{H})$, meaning that $V_{i}\supseteq UE_{i}$
for all $i\in I$. Note, that if $V$ corresponds to a frame
$(\varphi_{i})_{i\in I}$, i.e. each $V_{i}=\text{span}\\{\varphi_{i}\\}$ is
$1$-dimensional, then trivially $V_{i}=UE_{i}$ by Theorem 2.8, since we
exclude the case $V_{i}=\\{0\\}$ by default. In that sense, the next result is
indeed a generalization of the necessity-part of Theorem 2.8 (b). Note that in
case $V$ is a fusion Riesz basis, we have indeed $V_{i}=UE_{i}$ for all $i\in
I$, where $U\in\mathcal{B}(\mathcal{H})$ is bijective, see Proposition 8.

###### Proposition 7

Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion frame for $\mathcal{H}$. Then
$V_{i}\supseteq UE_{i}$ for all $i\in I$, where $(E_{i})_{i\in I}$ is an
orthonormal fusion basis and $U\in\mathcal{B}(\mathcal{H})$ is surjective.

###### Proof

Suppose that $(e_{ij})_{j\in J_{i}}$ is an orthonormal basis for $V_{i}$ for
every $i\in I$. Hence, according to Theorem 3.1, $(v_{i}e_{ij})_{i\in I,j\in
J_{i}}$ is a frame for $\mathcal{H}$. By Theorem 2.8, there exists a
surjective operator $U\in\mathcal{B}(\mathcal{H})$ and an orthonormal basis
$(\psi_{ij})_{i\in I,j\in J_{i}}$ for $\mathcal{H}$, such that
$v_{i}e_{ij}=U\psi_{ij}$ for all $i\in I,j\in J_{i}$. Set
$E_{i}:=\overline{\text{span}}(\psi_{ij})_{j\in J_{i}}$ for every $i\in I$.
Then $(E_{i})_{i\in I}$ is an orthonormal fusion basis for $\mathcal{H}$ and
we have $UE_{i}=U\overline{\text{span}}(\psi_{ij})_{j\in
J_{i}}\subseteq\overline{\text{span}}(U\psi_{ij})_{j\in
J_{i}}=\overline{\text{span}}(v_{i}e_{ij})_{j\in J_{i}}=V_{i}$ for every $i\in
I$, by Lemma 13.

###### Proposition 8

Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion Riesz basis for $\mathcal{H}$. Then
$(V_{i})_{i\in I}=(UE_{i})_{i\in I}$, where $(E_{i})_{i\in I}$ is an
orthonormal fusion basis and $U\in\mathcal{B}(\mathcal{H})$ is bijective.

###### Proof

Suppose that $(e_{ij})_{j\in J_{i}}$ is an orthonormal basis for $V_{i}$ for
every $i\in I$. By Theorem 4.3, $(v_{i}e_{ij})_{i\in I,j\in J_{i}}$ is a Riesz
basis for $\mathcal{H}$. Hence, by Theorem 2.8 (c), there exists a bijective
operator $U\in\mathcal{B}(\mathcal{H})$ and an orthonormal basis
$(\psi_{ij})_{i\in I,j\in J_{i}}$ for $\mathcal{H}$, such that
$v_{i}e_{ij}=U\psi_{ij}$ for all $i\in I,j\in J_{i}$. Set
$E_{i}:=\overline{\text{span}}(\psi_{ij})_{j\in J_{i}}$ for every $i\in I$.
Then $(E_{i})_{i\in I}$ is an orthonormal fusion basis for $\mathcal{H}$ and
we have $UE_{i}=U\overline{\text{span}}(\psi_{ij})_{j\in
J_{i}}=\overline{\text{span}}(U\psi_{ij})_{j\in
J_{i}}=\overline{\text{span}}(v_{i}e_{ij})_{j\in J_{i}}=V_{i}$ for every $i\in
I$, by Lemma 13.

We conclude this section with another major difference between properties of
frames and properties of fusion frames. Recall that if $(\varphi_{i})_{i\in
I}$ is a frame with frame operator $S_{\varphi}$, then the family
$(S_{\varphi}^{-1/2}\varphi_{i})_{i\in I}$ is always a Parseval frame ole1 .
In the fusion frame setting, the situation is very different. Below, we give
an example of a fusion frame $V=(V_{i},v_{i})_{i\in I}$, so that for any
arbitrary bounded and invertible operator $U\in\mathcal{B}(\mathcal{H})$,
$UV=(UV_{i},w_{i})_{i\in I}$ is _never_ a Parseval fusion frame, no matter
which family $(w_{i})_{i\in I}$ of weights we choose:

###### Example 4

Example rust08  Let $\mathcal{H}=\mathbb{C}^{4}$ and $(e_{i})_{i=1}^{4}$ be an
orthonormal basis for $\mathbb{C}^{4}$. Consider the fusion frame
$(V_{i},1)_{i=1}^{3}$ for $\mathbb{C}^{4}$, where
$V_{1}=\text{span}\\{e_{1},e_{2}\\}$, $V_{2}=\text{span}\\{e_{1},e_{3}\\}$ and
$V_{3}=\text{span}\\{e_{4}\\}$. We claim that for any invertible matrix
$U\in\mathbb{C}^{4\times 4}$ and any choice $(w_{i})_{i=1}^{3}$ of weights,
$(UV_{i},w_{i})_{i\in I}$ is never a Parseval fusion frame for
$\mathbb{C}^{4}$. Indeed, let $g_{1}=\|Ue_{1}\|^{-1}Ue_{1}$,
$g_{4}=\|Ue_{4}\|^{-1}Ue_{4}$ and choose $g_{2}$ and $g_{3}$ in such a way
that $(g_{1},g_{2})$ is an orthonormal basis for $UV_{1}$ and $(g_{1},g_{3})$
is an orthonormal basis for $UV_{2}$. Suppose that $(UV_{i},w_{i})_{i\in I}$
was a Parseval fusion frame. Then its associated global frame
$wg=(w_{1}g_{1},w_{2}g_{1},w_{1}g_{2},w_{2}g_{3},w_{3}g_{4})$ would be a
Parseval frame by Corollary 2. Let $D_{wg}\in\mathbb{C}^{4\times 5}$ be its
associated synthesis matrix, i.e. the matrix consisting of the frame elements
of $wg$ as column vectors. Further, let $H\in\mathbb{C}^{4\times 4}$ be a
unitary matrix whose first row is given by $g_{1}$ and denote the other rows
by $h_{2},h_{3},h_{4}$. Then

$H\cdot D_{wg}=\begin{bmatrix}w_{1}&w_{2}&\vec{v}\\\
\textbf{0}&\textbf{0}&A\\\ \end{bmatrix}$

for some $\vec{v}=(0,0,z)\in\mathbb{C}^{3}$ and $A\in\mathbb{C}^{3\times 3}$
given by

$A=\begin{bmatrix}\langle h_{2},w_{1}g_{2}\rangle&\langle
h_{2},w_{2}g_{3}\rangle&\langle h_{2},w_{3}g_{4}\rangle\\\ \langle
h_{3},w_{1}g_{2}\rangle&\langle h_{3},w_{2}g_{3}\rangle&\langle
h_{3},w_{3}g_{4}\rangle\\\ \langle h_{4},w_{1}g_{2}\rangle&\langle
h_{4},w_{2}g_{3}\rangle&\langle h_{4},w_{3}g_{4}\rangle\\\ \end{bmatrix}.$

Since
$HD_{wg}(HD_{wg})^{*}=HD_{wg}D_{wg}^{*}H^{*}=HS_{wg}H^{*}=HH^{*}=\mathcal{I}_{\mathbb{C}^{4}}$,
we must have $1=w_{1}^{2}+w_{2}^{2}+|z|^{2}$ and $A$ must be unitary. But this
is impossible because the first two columns of $A$ have norms
$\|w_{1}g_{2}\|=w_{1}$ and $\|w_{2}g_{3}\|=w_{2}$, while
$1=w_{1}^{2}+w_{2}^{2}+|z|^{2}$.

In particular, the above example demonstrates that for some fusion frames
$(V_{i},v_{i})_{i\in I}$, the fusion frame $(S_{V}^{-1/2}V_{i},w_{i})_{i\in
I}$ cannot be Parseval for any choice of weights $(w_{i})_{i\in I}$.

We can - though - formulate a particular positive result:

###### Proposition 9

Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion Riesz basis for $\mathcal{H}$. Then
$(S_{V}^{-1/2}V_{i})_{i\in I}$ is an orthonormal fusion basis for
$\mathcal{H}$.

###### Proof

Assume that $(e_{ij})_{j\in J_{i}}$ is an orthonormal basis for $V_{i}$ for
each $i\in I$. According to Theorem 4.3 and Lemma 11, $(v_{i}e_{ij})_{j\in
J_{i},i\in I}$ is a Riesz basis for $\mathcal{H}$ with the frame operator
$S_{V}$. Therefore $(v_{i}S_{V}^{-1/2}e_{ij})_{j\in J_{i},i\in I}$ is an
orthonormal basis for $\mathcal{H}$. Combined with the fact that
$(v_{i}S_{V}^{-1/2}e_{ij})_{j\in J_{i}}$ is an orthonormal basis of
$S_{V}^{-1/2}V_{i}$ for each $i\in I$, we see that $(S_{V}^{-1/2}V_{i})_{i\in
I}$ is an orthonormal fusion basis for $\mathcal{H}$.

##  6 Duality in fusion frame theory

In this section we present the duality theory for fusion frames. The
difference between frames and fusion frames becomes more evident in duality
theory. Hence we decided to first show the origin of the concept of dual
fusion frames. We then present the duality theory for fusion frames following
heimoanza14 ; hemo14 . We also adapt results of HeinekenMorillas18 , where the
concept is analyzed in the more general setting of oblique duality. We study
properties and provide examples of this concept. Finally, we will discuss
particular cases and other approaches for a dual fusion frame concept.

Recall, that in frame theory each $f\in\mathcal{H}$ is represented by the
collection of scalar coefficients $\langle f,\varphi_{i}\rangle$, $i\in I$,
that can be thought as a measure of the projection of $f$ onto each frame
vector. From these coefficients $f$ can be stably recovered using a
reconstruction formula via dual frames as in (5), i.e.

$\sum_{i\in I}\langle f,\psi_{i}\rangle\varphi_{i}=\sum_{i\in I}\langle
f,\varphi_{i}\rangle\psi_{i}=f\qquad(\forall f\in\mathcal{H}).$

In operator notation, this reads

$D_{\varphi}C_{\psi}=D_{\psi}C_{\varphi}=\mathcal{I}_{\mathcal{H}}.$ (26)

As already mentioned in Section 2.1, a redundant frame has infinitely many
dual frames, which makes frames so desirable for countless applications.
Taking this into account, our aim is to have a notion of a dual fusion frame
as we have it in classical frame theory, and that furthermore leads to
analogous results. Since the duality condition can be expressed in the two
forms (5) and (26), it is natural to try to generalize these expressions to
the context of fusion frames in order to obtain a definition of dual fusion
frame.

The concept that satisfies the properties which are desirable for a dual
fusion frame came up and was studied in heimoanza14 ; hemo14 , considering as
coefficient space $\mathcal{K}_{V}^{2}$. The reasoning there was the
following:

Let $V=(V_{i},v_{i})_{i\in I}$ be a fusion frame. Since
$S^{-1}_{V}S_{V}=\mathcal{I}_{\mathcal{H}}$, we have the following
reconstruction formula

$f=\sum_{i\in I}v_{i}^{2}S_{V}^{-1}\pi_{V_{i}}f\qquad(\forall
f\in\mathcal{H}),$ (27)

which is analogous to (4). The family $(S_{V}^{-1}V_{i},v_{i})_{i\in I}$ is a
fusion frame (see Corollary 9) which in (caskut04, , Definition 3.19) is
called the dual fusion frame of $V$, and is similar to the canonical dual
frame in the classical frame theory. As it is pointed out in (CK2, , Chapter
13), (27) - in contrast to (5) for frames - does not lead automatically to a
dual fusion frame concept.

So, instead of trying to generalize (5), we can try with (26). But in this
case we find the following obstacle. Given two fusion frames
$V=(V_{i},v_{i})_{i\in I}$ and $W=(W_{i},w_{i})_{i\in I}$ for $\mathcal{H}$,
with $(V_{i})_{i\in I}\neq(W_{i})_{i\in I}$, the corresponding synthesis
operators $D_{V}$ and $D_{W}$ have different domains if defined - as in the
original papers - on the respective subspaces $\mathcal{K}_{V}^{2}$ and
$\mathcal{K}_{W}^{2}$ of $\mathcal{K}_{\mathcal{H}}^{2}$. Therefore the
composition of $D_{W}$ with $C_{V}$ is not possible. In heimoanza14  (see also
hemo14 ), to overcome this domain issue,
$Q:\mathcal{K}_{V}^{2}\longrightarrow\mathcal{K}_{W}^{2}$ is inserted between
$D_{W}$ and $C_{V}$:

$D_{W}QC_{V}$

giving origin to the commonly named $Q$-dual fusion frames. The philosophy
then was to require additional conditions on $Q$, in order to obtain the
different desired properties for dual fusion frames, similar to those which
exist for dual frames. This is why first a general $Q$ is used, asking only
for boundedness of $Q$, getting with this minimal restriction
characterizations analogous to those that exist for dual frames (see Section
6.1).

In order to have computationally convenient reconstruction formulae, the
_block-diagonal dual fusion frames_ were introduced in hemo14  (see also
HeinekenMorillas18 ). Dual fusion frame systems hemo14 ; HeinekenMorillas18 ,
which are a useful tool for local processing tasks, turned out to be block-
diagonal too. Studying the relation between dual fusion frames and the bounded
left-inverses of the analysis operator, a particular class of block-diagonal
dual fusion frames, called component preserving dual fusion frames aroused
HeinekenMorillas18 ; heimoanza14 ; hemo14 . They can be obtained from the left
inverse of the analysis operator and hence are easy to construct. These are of
particular importance, since the canonical dual fusion frame is of this type.
Moreover, they are crucial in the applications, since they are optimal in the
presence of erasures hemo14 ; Mo17  (see Section 6.2).

If one considers $\mathcal{K}_{\mathcal{H}}^{2}$ as coefficient space $-$ as
we have done in this survey so far $-$ there is no domain problem in the
composition of $D_{W}$ with $C_{V}$ and hence there is in principle no need to
include an operator $Q$ between them. In this case, the first idea, which
probably comes to mind when searching for a dual fusion frame definition, is
generalizing the duality concept directly from (26). However, this approach
does not lead to a completely satisfactory notion of duality (see Section