# Vector-Valued (Super) Weaving Frames

Deepshikha  Deepshikha, Department of Mathematics, University of Delhi,
Delhi-110007, India<EMAIL_ADDRESS>and  Lalit Kumar Vashisht  Lalit K.
Vashisht, Department of Mathematics, University of Delhi, Delhi-110007, India
<EMAIL_ADDRESS>
###### Abstract.

Two frames $\\{\phi_{i}\\}_{i\in I}$ and $\\{\psi_{i}\\}_{i\in I}$ for a
separable Hilbert space $H$ are woven if there are positive constants $A\leq
B$ such that for every subset $\sigma\subset I$, the family
$\\{\phi_{i}\\}_{i\in\sigma}\cup\\{\psi_{i}\\}_{i\in\sigma^{c}}$ is a frame
for $H$ with frame bounds $A,B$. Bemrose et al. introduced weaving frames in
separable Hilbert spaces and observed that weaving frames has potential
applications in signal processing. Motivated by this, and the recent work of
Balan in the direction of application of vector-valued frames (or superframes)
in signal processing, we study vector-valued weaving frames. In this paper,
first we give some fundamental properties of vector-valued weaving frames. It
is shown that if a family of vector-valued frames is woven, then the
corresponding family of frames for atomic spaces is woven, but the converse is
not true. We present a technique for the construction of vector-valued woven
frames from given woven frames for atomic spaces . Necessary and sufficient
conditions for vector-valued weaving Riesz sequences are given. Several
numerical examples are given to illustrate the results.

###### Key words and phrases:

Frames, vector-valued frames, weaving frames, Riesz basis.
The research of Deepshikha is supported by CSIR, India vide File No.:
09/045(1352)/2014-EMR-I. Lalit was partially supported by R $\&$ D Doctoral
Research Programme, University of Delhi, Delhi-110007, India (Grant No.:
RC/2015/9677).

###### 2010 Mathematics Subject Classification:

Primary 42C15 Secondary 42C30, 42C40.

##  1\. Introduction

Duffin and Schaeffer [12] introduced frames for Hilbert spaces in the context
of nonharmonic Fourier series. Today frames have applications in a wide range
of areas in applied mathematics. The applications of frames in signal
processing are now well-known, for example see [6, 7]. Balan [1] introduced
the concept of a vector-valued frame (or “superframe”) in the context of
multiplexing and further studied in [2]. The vector-valued frame has
significant applications in mobile communication, satellite communication, and
computer area network. Recently, Bemrose, Casazza, Gröchenig, Lammers and
Lynch in [3] proposed weaving frames in a separable Hilbert space. The concept
of weaving frames is motivated by a problem regarding distributed signal
processing where redundant building blocks (frames) plays an important role.
For example, in wireless sensor networks where frames may be subjected to
distributed processing under different frames. Motivated by the concept of
weaving frames and superframes and their application in Gabor and wavelet
analysis, in this paper, we study vector-valued (super) weaving frames.
Weaving frames has potential applications in wireless sensor networks that
require distributed processing under different frames, as well as
preprocessing of signals using Gabor frames. Notable contribution in the paper
is a new technique for the construction of vector-valued weaving frames from
frames of atomic spaces. Some necessary and sufficient conditions for vector-
valued weaving Riesz sequences are given. Finally, a result for vector-valued
weaving Riesz sequences in terms of operators on atomic spaces has been
obtained.

###  1.1. Previous works on weaving frames

For a positive integer $m$, we write $[m]=\\{1,2,\dots,m\\}$. We start with
the definition of weaving frames in separable Hilbert spaces.

######  Definition 1.1.

[3] A family of frames $\\{\phi_{ij}\\}_{i\in I}$ for $j\in[m]$ for a Hilbert
space $H$ is said to be _woven_ if there are universal constants $A$ and $B$,
so that for every partition $\\{\sigma_{j}\\}_{j\in[m]}$ of $I$, the family
$\\{\phi_{ij}\\}_{i\in\sigma_{j},j\in[m]}$ is a frame for $H$ with lower and
upper frame bounds $A$ and $B$, respectively.

######  Definition 1.2.

[3] A family of frames $\\{\phi_{ij}\\}_{i\in\mathbb{N},j\in[m]}$ for a
Hilbert space $H$ is said to be _weakly woven_ if for every partition
$\\{\sigma_{j}\\}_{j\in[m]}$ of $\mathbb{N}$, the family
$\\{\phi_{ij}\\}_{i\in\sigma_{j},j\in[m]}$ is a frame for $H$.

######  Remark 1.3.

It is proved in [3] that this weaker form of weaving (given in Definition 1.2)
is equivalent to weaving

Bemrose et al. proved in [3] that a frame (which is not a Riesz basis) cannot
be weave with a Riesz basis.

######  Theorem 1.4.

[3] Let $\\{\phi_{j}\\}_{j=1}^{\infty}$ be a Riesz basis and let
$\\{\psi_{j}\\}_{j=1}^{\infty}$ be a frame for $H$. If
$\\{\phi_{j}\\}_{j=1}^{\infty}$ and $\\{\psi_{j}\\}_{j=1}^{\infty}$ are woven,
then $\\{\psi_{j}\\}_{j=1}^{\infty}$ must actually be a Riesz basis.

Bemrose et al. in [3] classified when Riesz bases and Riesz basic sequences
can be woven and provide a characterization in terms of distances between
subspaces. Furthermore, they proved that if two Riesz bases are woven, then
every weaving is in fact a Riesz basis, and not just a frame. A geometric
characterization of woven Riesz bases in terms of distance between subspaces
of a Hilbert space $H$ is given in [3]. Casazza and Lynch in [4] reviewed
fundamental properties of weaving frames. They considered a relation of frames
to projections and gave a better understanding of what it really means for two
frames to be woven. Finally, they discussed a weaving equivalent of an
unconditional basis. Generalized weaving frames in separable Hilbert spaces
were studied in [8, 10, 23]. Some necessary and sufficient conditions for
weaving fusion frames may found in [9]. In [24, 25], Vashisht and Deepshikha
studied weaving frames with respect to measure spaces. Dörfler and Faulhuber
[13] studied weaving Gabor frames in $L^{2}(\mathbb{R})$. They also discussed
a family of localization operators related to weaving Gabor frames. A
sufficient criteria for a family of multi-window Gabor frames to be woven may
found in [13]. Casazza, Freeman and Lynch [5] extended the concept of weaving
Hilbert space frames to the Banach space setting. They introduced and studied
_weaving Schauder frames_ in Banach spaces.

###  1.2. Overview and Main Results

The paper is organized as follows: Section 2 contains the basic definitions
and results about frames, weaving frames and vector-valued frames. In Section
3, we introduce vector-valued weaving frames. Some properties of vector-valued
frames are given. It is shown that frames associated with atomic spaces are
woven, provided the given family of vector-valued frames is woven, see
Proposition 3.5. The converse is not true. Theorem 3.9 gives a technique for
the construction of vector-valued woven frames from woven frames for atomic
spaces. In Section 4, we give necessary and sufficient conditions for vector-
valued weaving Riesz sequences. Proposition 4.4 gives a sufficient condition
for vector-valued weaving Riesz bases. It is shown in Theorem 4.7 that if a
family of frames associated with one of the atomic spaces is a woven Riesz
sequence, then the family of vector-valued frames is also a woven Riesz
sequence. Finally, sufficient conditions for vector-valued weaving Riesz
sequences in terms of bounded operators on atomic spaces are given in Theorem
4.9. Several examples and counter-example are given to illustrate our results.

##  2\. Preliminaries

In this section, we review the concepts of frames, vector-valued frames and
weaving frames. We begin with some notations: The set of all positive integers
is denoted by $\mathbb{N}$ or $\mathbb{Z}^{+}$. By $\mathbb{Z}$ and
$\mathbb{Z}^{-}$ we denote the set of all integers and negative integers,
respectively and $\mathbb{Z}^{*}=\mathbb{Z}\setminus\\{0\\}$. For a set $E$,
$\chi_{E}$ denote the characteristic function of $E$. By $\mathbb{K}$ we
denote field of real or complex numbers. For an indexing set $I$, as usual
$\ell^{2}(I)$ is the space of all square summable complex-valued sequences
indexed by the set $I$. The space of bounded linear operators from a normed
space $X$ into a normed space $Y$ is denoted by $\mathcal{B}(X,Y)$. If $X=Y$,
then we write $\mathcal{B}(X,Y)=\mathcal{B}(X)$.

###  2.1. Hilbert space frames:

A countable sequence $\\{f_{k}\\}_{k\in I}$ in a separable Hilbert space $H$
is called a _frame_ (or _Hilbert frame_) for $H$ if there exist positive
numbers $\alpha_{o}\leq\beta_{o}<\infty$ such that

$\displaystyle\alpha_{o}\|f\|^{2}\leq\sum_{k\in I}|\langle
f,f_{k}\rangle|^{2}\leq\beta_{o}\|f\|^{2}\ \text{for all}\ f\in H.$ (2.1)

The numbers $\alpha_{o}$ and $\beta_{o}$ are called _lower_ and _upper frame
bounds_ , respectively. If upper inequality in (2.1) is satisfied, then we say
that $\\{f_{k}\\}_{k\in I}$ is a _Bessel sequence_ (or _Hilbert Bessel
sequence_) with _Bessel bound_ $\beta_{o}$. The frame $\\{f_{k}\\}_{k\in I}$
is _tight_ if it is possible to choose $\alpha_{o}=\beta_{o}$.

Let $\\{f_{k}\\}_{k\in I}$ be a frame (or a Bessel sequence) for $H$. The
_pre-frame operator_ operator is a linear bounded operator
$T:\ell^{2}(I)\rightarrow H$ given by

$\displaystyle T\\{c_{k}\\}_{k\in I}=\sum_{k\in I}c_{k}f_{k}.$

The frame operator $S=TT^{*}:H\rightarrow H$ is

$\displaystyle Sf=\sum_{k\in I}\langle f,f_{k}\rangle f_{k}.$

The frame operator for a frame $\\{f_{k}\\}_{k\in I}$ is bounded linear and
invertible operator on $H$. This gives the _reconstruction formula_ for all
$f\in H$,

$\displaystyle f=SS^{-1}f=\sum_{k\in I}\langle S^{-1}f,f_{k}\rangle f_{k}\
=\sum_{k\in I}\langle f,S^{-1}f_{k}\rangle f_{k}.$

The scalars $\\{\langle S^{-1}f,f_{k}\rangle\\}_{k\in I}$ are called _frame
coefficients_ of the vector $f\in H$. One may observe that the representation
of $f$ in the reconstruction formula need not be unique.

A _Riesz basis_ for $H$ is a family of the form $\\{Ue_{k}\\}_{k\in I}$, where
$\\{e_{k}\\}_{k\in I}$ is an orthonormal basis for $H$ and $U:H\rightarrow H$
is a bounded linear bijective operator. The following result for bases in
Banach spaces can be found in [20, p. 173]

######  Theorem 2.1.

[20] Let $\\{x_{n}\\}_{n\in\mathbb{N}}$ be a basis for a Banach space $X$. If
$\\{y_{n}\\}_{n\in\mathbb{N}}\subset X$ and there exists a constant
$\lambda\in[0,1)$ such that

$\displaystyle\Big{\|}\sum_{n=1}^{N}c_{n}(x_{n}-y_{n})\Big{\|}\leq\lambda\
\Big{\|}\sum_{n=1}^{N}c_{n}x_{n}\Big{\|},\ N\in\mathbb{N},\ \
c_{1},c_{2},\dots,c_{N}\in\mathbb{K}.$

then $\\{y_{n}\\}_{n\in\mathbb{N}}$ is a basis for $X$ and
$\\{y_{n}\\}_{n\in\mathbb{N}}$ is equivalent to
$\\{x_{n}\\}_{n\in\mathbb{N}}$. That is, there exists a bounded, linear and
bijective operator $T:X\rightarrow X$ such that $Tx_{n}=y_{n}$ for all
$n\in\mathbb{N}$.

The basic theory of frames can be found in books of Casazza and Kutyniok [6]
and Christensen [7].

###  2.2. Vector-valued Franes (Superframes):

Let $I$ be a countable index set and consider
$(\mathcal{F}_{1};\pi_{1};I),\dots,(\mathcal{F}_{L};\pi_{L};I)$, $L$ indexed
sets of vectors (not necessarily from the same Hilbert space), where
$\pi_{k}:I\rightarrow F_{k}$ is the corresponding indexing map. A collection
of such countable sets of vectors together with their corresponding indexing
maps from a same index set is called a _superset_. In short, we write
$(F_{1},....,F_{L})$ for a superset when an indexing by a same index set $I$
for each subset $F_{k}$ of vectors of some Hilbert space $\mathcal{H}_{k}$ (or
a bigger space $K_{k})$ is fixed.
We write

$\displaystyle\mathcal{F}=\mathcal{F}_{1}\oplus\cdots\oplus\mathcal{F}_{L}=\\{f^{1}_{i}\oplus\cdots\oplus
f_{i}^{L}:i\in I\\},\ f^{k}_{i}=\pi_{k}(i)\in\mathcal{F}_{k}.$

Recall that the space $\mathcal{H}_{1}\oplus....\oplus\mathcal{H}_{L}$ is a
Hilbert space with natural inner product:

$\displaystyle\langle f_{1}\oplus\cdots\oplus f_{L},\ g_{1}\oplus\cdots\oplus
g_{L}\rangle=\sum_{i=1}^{L}\langle f_{i},g_{i}\rangle_{\mathcal{H}_{i}},\
\Big{(}f_{i},g_{i}\in\mathcal{H}_{i}\ (1\leq i\leq L)\Big{)}.$

######  Definition 2.2.

[1] The superset $(F_{1},\dots,F_{L})$ is called a _vector-valued frame_ or
_super frame_ if $\mathcal{F}$ is a frame for the space
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$. That is, if there exists
finite positive numbers $A_{0}\leq B_{0}$ such that for every
$h_{k}\in\mathcal{H}_{k}\ (1\leq k\leq L)$, we have

$\displaystyle A_{0}(\|h_{1}\|^{2}+\cdots+\|h_{L}\|^{2})\leq\sum_{i\in
I}\Big{|}\sum_{k=1}^{L}\langle h_{k},f^{k}_{i}\rangle\Big{|}^{2}\leq
B_{0}(\|h_{1}\|^{2}+\cdots+\|h_{L}\|^{2}).$

If $I$ is the common indexing set (possibly countably infinite) and
$F_{i}=\\{f_{ik}\\}_{k\in I},\ i=1,2,\dots,L$, then a vector-valued family is
represented by $\\{F_{1},\dots,F_{L},I\\}$. The spaces $\mathcal{H}_{i}$ are
called _atomic spaces_ (or _components_) of the space
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$.

By using superframes, Balan in [1] discussed a signal transmission problem
(namely, multiplexing, or Multiple Access) and analyzed several encoding-
decoding schemes (with an example). In recent years, vector-valued wavelet and
Gabor frames have attracted mathematician and engineering specialists, for
example see [14, 15, 16, 17, 19, 21, 22] and reference therein.

##  3\. Vector-Valued Weaving Frames

We start this section by the definition of a vector-valued weaving frame.

######  Definition 3.1.

A family of vector-valued frames
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$ for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ is said to be _woven_ if
there exist universal constants $A$ and $B$ such that for any partition
$\\{\sigma_{i}\\}_{i\in[m]}$ of $I$, the family

$\displaystyle\bigcup_{i\in[m]}\\{F_{1}^{i},\dots,F_{L}^{i},\sigma_{i}\\}$

is a vector-valued frame for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ with lower and upper frame
bounds $A$ and $B$, respectively. Each family
$\bigcup_{i\in[m]}\\{F_{1}^{i},\dots,F_{L}^{i},\sigma_{i}\\}$ is called a
_vector-valued weaving_ (or simply _weaving_).

######  Remark 3.2.

Throughout the paper we take $F_{j}\subset\mathcal{H}_{j}$ for $j\in[L]$. If
$\\{F_{1},\dots,F_{L},I\\}$ is vector-valued frame (Bessel sequence) for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$, then $F_{j}\
(j=1,\dots,L)$ are frames (Bessel sequences) for atomic spaces
$\mathcal{H}_{j}$.

Regarding the existence of vector-valued weaving frames, we have the following
examples.

######  Example 3.3.

Let $L=m=2;\ I=\mathbb{N}$ and let
$\mathcal{H}_{1}=\mathcal{H}_{2}=\ell^{2}(\mathbb{N})$.
Define $F_{1}=\\{f_{1i}\\}_{i\in I},G_{1}=\\{g_{1i}\\}_{i\in
I}\subset\mathcal{H}_{1}$ as follows:

$\displaystyle f_{1i}=$ $\displaystyle\begin{cases}e_{j},\ i=4j-3\
(j\in\mathbb{N})\\\ 0,\ \text{otherwise},\end{cases}\quad\text{and}\quad
g_{1i}=\begin{cases}e_{j},\ i=4j-3,4j-1\ (j\in\mathbb{N})\\\ 0,\
\text{otherwise}.\end{cases}$

and $F_{2}=\\{f_{2i}\\}_{i\in I},G_{2}=\\{g_{2i}\\}_{i\in
I}\subset\mathcal{H}_{2}$ as follows:

$\displaystyle f_{2i}=$ $\displaystyle\begin{cases}e_{j},\ i=4j\
(j\in\mathbb{N})\\\ 0\ \text{otherwise},\end{cases}\quad\text{and}\quad
g_{2i}=\begin{cases}e_{j},\ i=4j-2,4j\ (j\in\mathbb{N})\\\ 0\
\text{otherwise},\end{cases}$

where $\\{e_{i}\\}_{i\in I}$ is the canonical orthonormal basis of
$\ell^{2}(\mathbb{N}$.

To show the vector-valued families $\\{F_{1},F_{2},I\\}$ and
$\\{G_{1},G_{2},I\\}$ are woven, let $\sigma\subset I$ be any subset.
We compute

$\displaystyle\sum_{i\in\sigma}\Big{|}\Big{\langle}f\oplus g,f_{1i}\oplus
f_{2i}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}}\Big{|}\Big{\langle}f\oplus
g,g_{1i}\oplus g_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle\geq\sum_{i\in\sigma\bigcap\left(\\{4m-3\\}_{m\in\mathbb{N}}\cup\\{4m\\}_{m\in\mathbb{N}}\right)}\Big{|}\Big{\langle}f\oplus
g,f_{1i}\oplus f_{2i}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}\
\bigcap\left(\\{4m-3\\}_{m\in\mathbb{N}}\cup\\{4m\\}_{m\in\mathbb{N}}\right)}\Big{|}\Big{\langle}f\oplus
g,g_{1i}\oplus g_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle=\Big{|}\Big{\langle}f\oplus g,e_{1}\oplus
0\Big{\rangle}\Big{|}^{2}+\Big{|}\Big{\langle}f\oplus g,0\oplus
e_{1}\Big{\rangle}\Big{|}^{2}+\Big{|}\Big{\langle}f\oplus g,e_{2}\oplus
0\Big{\rangle}\Big{|}^{2}+\Big{|}\Big{\langle}f\oplus g,0\oplus
e_{2}\Big{\rangle}\Big{|}^{2}+\cdots$ $\displaystyle=\Big{|}\langle
f,e_{1}\rangle\Big{|}^{2}+\big{|}\langle
g,e_{1}\rangle\Big{|}^{2}+\Big{|}\langle
f,e_{2}\rangle\Big{|}^{2}+\Big{|}\langle g,e_{2}\rangle\Big{|}^{2}+\cdots$
$\displaystyle=\|f\|^{2}+\|g\|^{2}$ $\displaystyle=\|f\oplus g\|^{2}\
\text{for all}\ f\oplus g\in\mathcal{H}_{1}\oplus\mathcal{H}_{2}.$

This gives a lower universal bound $A=1$. Similarly, we can show that an upper
universal bound is $B=2$. Hence the vector-valued frames $\\{F_{1},F_{2},I\\}$
and $\\{G_{1},G_{2},I\\}$ for $\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ are
woven.

######  Example 3.4.

Let $L,m,I,\ \mathcal{H}_{1}$ and $\mathcal{H}_{2}$ be same as in Example 3.3.

Define $F_{1}=\\{f_{1i}\\}_{i\in I},G_{1}=\\{g_{1i}\\}_{i\in
I}\subset\mathcal{H}_{1}$ as follows:

$\displaystyle f_{1i}=$ $\displaystyle\begin{cases}e_{j},\ i=6j\
(j\in\mathbb{N})\\\ 0,\ \text{otherwise}.\end{cases}\quad\text{and}\quad
g_{1i}=\begin{cases}e_{j},\ i=6j-3,6j-1,6j\ (j\in\mathbb{N})\\\ 0,\
\text{otherwise}.\end{cases}$

and $F_{2}=\\{f_{2i}\\}_{i\in I},G_{2}=\\{g_{2i}\\}_{i\in
I}\subset\mathcal{H}_{2}$ as follows:

$\displaystyle f_{2i}=$ $\displaystyle\begin{cases}e_{j},\ i=6j-1\
(j\in\mathbb{N})\\\ 0\ \text{otherwise}.\end{cases}\quad\text{and}\quad
g_{2i}=\begin{cases}e_{j},\ i=6j-4,6j-1,6j\ (j\in\mathbb{N})\\\ 0\
\text{otherwise}.\end{cases}$

where $\\{e_{i}\\}_{i\in I}$ is the canonical orthonormal basis of
$\mathcal{H}_{1}$. Then, $\\{F_{1},F_{2},I\\}$ and $\\{G_{1},G_{2},I\\}$ are
vector-valued frames for $\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ but not woven.

Indeed, suppose $\\{F_{1},F_{2},I\\}$ and $\\{G_{1},G_{2},I\\}$ are woven with
universal bounds $A$ and $B$.
For $\sigma=I\setminus\\{5,6\\}$ and
$\left(-1,0,0,0,\dots\right)\oplus\left(1,0,0,0,\dots\right)\in\mathcal{H}_{1}\oplus\mathcal{H}_{2}$,
we compute

$\displaystyle\sum_{i\in\sigma}\Big{|}\Big{\langle}\left(-1,0,0,0,\dots\right),f_{1i}\Big{\rangle}+\Big{\langle}\left(1,0,0,0,\dots\right),f_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle\quad\quad\quad\quad\quad+\sum_{i\in\sigma^{c}}\Big{|}\Big{\langle}\left(-1,0,0,0,\dots\right),g_{1i}\Big{\rangle}+\Big{\langle}\left(1,0,0,0,\dots\right),g_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle=\Big{|}\Big{\langle}\left(-1,0,0,0,\dots\right),e_{1}\Big{\rangle}+\Big{\langle}\left(1,0,0,0,\dots\right),e_{1}\Big{\rangle}\Big{|}^{2}+\Big{|}\Big{\langle}\left(-1,0,0,0,\dots\right),e_{1}\Big{\rangle}+\Big{\langle}\left(1,0,0,0,\dots\right),e_{1}\Big{\rangle}\Big{|}^{2}$
$\displaystyle=0$
$\displaystyle<2A=A\Big{(}\Big{\|}(-1,0,0,0,\dots)\oplus(1,0,0,0,\dots)\Big{\|}^{2}\Big{)},\
\text{a contradiction}.$

Hence the vector-valued frames $\\{F_{1},F_{2},I\\}$ and $\\{G_{1},G_{2},I\\}$
for $\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ are not woven.

The following proposition shows that if a family of vector-valued frames is
woven, then it is componentwise woven. That is, frames associated with atomic
spaces are woven. The converse is not true, see Example 3.6.

######  Proposition 3.5.

Suppose the family
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$ of vector-valued
frames for $\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ is woven with
universal bounds $A$ and $B$. Then, for each $j\in[L]$, the frames
$\Big{\\{}F_{j}^{i}:i\in[m]\Big{\\}}$ for $\mathcal{H}_{j}$ are woven.

###### Proof.

Let $j\in[L]$ be arbitrary but fixed and let $\\{\sigma_{i}\\}_{i\in[m]}$ be
any partition of $I$. Let us write $F_{p}^{i}=\\{f_{pk}^{i}\\}_{k\in I}$,
$p\in[L],\ i\in[m]$.

Then, for any $f\in\mathcal{H}_{j}$ $($note that $0\oplus\cdots\oplus
f\oplus\cdots\oplus
0\in\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{j}\oplus\cdots\oplus\mathcal{H}_{L})$,
we compute

$\displaystyle A\|f\|^{2}$
$\displaystyle\leq\sum_{i\in[m]}\sum_{k\in\sigma_{i}}\Big{|}\langle
0,f_{1k}^{i}\rangle+\cdots+\langle f,f_{jk}^{i}\rangle+\cdots+\langle
0,f_{Lk}^{i}\rangle\Big{|}^{2}$
$\displaystyle=\sum_{i\in[m]}\sum_{k\in\sigma_{i}}\Big{|}\langle
f,f_{jk}^{i}\rangle\Big{|}^{2}$ $\displaystyle\leq B\|f\|^{2}.$

Hence the family of frames $\Big{\\{}F_{j}^{i}:i\in[m]\Big{\\}}$ for the
atomic space $\mathcal{H}_{j}$ is woven. ∎

We now demonstrate by a concrete example that converse of Proposition 3.5 is
not true.

######  Example 3.6.

Let $F_{1},G_{1}\subset\mathcal{H}_{1}$ and
$F_{2},G_{2}\subset\mathcal{H}_{2}$ be frames for $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$, respectively given in Example 3.4. Then, for any subset
$\sigma$ of $I$, the families
$\\{f_{1i}\\}_{i\in\sigma}\bigcup\\{g_{1i}\\}_{i\in\sigma^{c}}$ and
$\\{f_{2i}\\}_{i\in\sigma}\bigcup\\{g_{2i}\\}_{i\in\sigma^{c}}$ are frames for
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, respectively with frame bounds $1$
and $3$. That is, the frames $F_{i}$ and $G_{i}$ for $\mathcal{H}_{i}$
$(i=1,2)$ are woven. But as shown in Example 3.4, the vector-valued frames
$\\{F_{1},F_{2},I\\}$ and $\\{G_{1},G_{2},I\\}$ are not woven.

However, this is not the case for Bessel sequences.

######  Proposition 3.7.

Suppose that $\\{F_{1}^{i},\dots,F_{L}^{i},I\\}$ are vector-valued families
for $\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$, where
$F_{k}^{i}=\\{f_{kj}^{i}\\}_{j\in I}$ $(k\in[L],i\in[m])$ such that
$\Big{\\{}\\{f_{kj}^{i}\\}_{j\in I}:i\in[m]\Big{\\}}$ is a woven Bessel
sequence, for all $k\in[L]$ . Then, the family
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$ is a woven
vector valued Bessel sequence for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$.

###### Proof.

Let $\Big{\\{}\\{f_{kj}^{i}\\}_{j\in I}:i\in[m]\Big{\\}}$ be a woven Bessel
sequence with universal bound $B_{k}$, for all $k\in[L]$. Then, for any
partition $\\{\sigma_{i}\\}_{i\in[m]}$ of $I$ and for any
$(h_{1},\dots,h_{L})\in\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$, we
have

$\displaystyle\sum_{i\in[m]}\sum_{j\in\sigma_{i}}\Big{|}\sum_{k=1}^{L}\langle
h_{k},f_{kj}^{i}\rangle\Big{|}^{2}$
$\displaystyle\leq\sum_{i\in[m]}\sum_{j\in\sigma_{i}}\sum_{k=1}^{L}2^{L-1}\Big{|}\langle
h_{k},f_{kj}^{i}\rangle\Big{|}^{2}$
$\displaystyle=\sum_{k=1}^{L}2^{L-1}\left(\sum_{i\in[m]}\sum_{j\in\sigma_{i}}\Big{|}\langle
h_{k},f_{kj}^{i}\rangle\Big{|}^{2}\right)$
$\displaystyle\leq\sum_{k=1}^{L}2^{L-1}B_{k}\|h_{k}\|^{2}$ $\displaystyle\leq
2^{L-1}\max_{k\in[L]}\\{B_{k}\\}\Big{(}\|h_{1}\|^{2}+\cdots+\|h_{L}\|^{2}\Big{)}.$

This concludes the proof. ∎

######  Corollary 3.8.

Suppose that $\\{F_{1}^{i},\dots,F_{L}^{i},I\\}$ are vector-valued families
for $\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$, where
$F_{k}^{i}=\\{f_{kj}^{i}\\}_{j\in I}$ is a Bessel sequence for all
$k\in[L],i\in[m]$. Then,
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$ is a woven
vector-valued Bessel sequence for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$.

A natural question arises about the construction of vector-valued woven frames
from given woven frames for atomic spaces. In this direction, the following
theorem provides a technique for the construction of a family of vector-valued
woven frames from given woven frames for atomic spaces. We prove the result
for $L=2$. This can be extended to any finite superset.

######  Theorem 3.9.

Assume that $\Phi_{j}=\\{\phi_{i}^{j}\\}_{i\in I_{j}}$ and
$\Psi_{j}=\\{\psi_{i}^{j}\\}_{i\in I_{j}}$ are woven frames for the Hilbert
space $\mathcal{H}_{j}$ $(j=1,2)$. Then, there is a vector-valued family
associated with $\Phi_{j}$ and $\Psi_{j}$ $(j=1,2)$ which constitutes vector-
valued woven frames for $\mathcal{H}_{1}\oplus\mathcal{H}_{2}$.

###### Proof.

Since $I_{1}$ and $I_{2}$ are countably infinite sets, there exist bijective
maps $\Theta_{1}:\mathbb{Z}^{-}\rightarrow I_{1}$ and
$\Theta_{2}:\mathbb{Z}^{+}\rightarrow I_{2}$.
Define $F_{1}=\\{f_{1i}\\}_{i\in\mathbb{Z}^{*}}\subset\mathcal{H}_{1}$ and
$F_{2}=\\{f_{2i}\\}_{i\in\mathbb{Z}^{*}}\subset\mathcal{H}_{2}$ as follows:

$\displaystyle f_{1i}=$ $\displaystyle\begin{cases}\phi_{\Theta_{1}(i)}^{1},\
i\in\mathbb{Z}^{-}\\\ 0,\ i\in\mathbb{Z}^{+},\end{cases}\quad\text{and}\quad
f_{2i}=\begin{cases}0,\ i\in\mathbb{Z}^{-}\\\ \phi_{\Theta_{2}(i)}^{2},\
i\in\mathbb{Z}^{+}.\end{cases}$

and define $G_{1}=\\{g_{1i}\\}_{i\in\mathbb{Z}^{*}}\subset\mathcal{H}_{1}$ and
$G_{2}=\\{g_{2i}\\}_{i\in\mathbb{Z}^{*}}\subset\mathcal{H}_{2}$ by

$\displaystyle g_{1i}=$ $\displaystyle\begin{cases}\psi_{\Theta_{1}(i)}^{1},\
i\in\mathbb{Z}^{-}\\\ 0,\ i\in\mathbb{Z}^{+}\end{cases}\quad\text{and}\quad
g_{2i}=\begin{cases}0,\ i\in\mathbb{Z}^{-}\\\ \psi_{\Theta_{2}(i)}^{2},\
i\in\mathbb{Z}^{+},\end{cases}$

To show that the vector-valued family $\\{F_{1},F_{2},\mathbb{Z}^{*}\\}$ and
$\\{G_{1},G_{2},\mathbb{Z}^{*}\\}$ are woven, let $\sigma$ be any subset of
$\mathbb{Z}^{*}$ and $f\oplus g\in\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ be
arbitrary.
We compute

$\displaystyle\sum_{i\in\sigma}\Big{|}\Big{\langle}f\oplus g,f_{1i}\oplus
f_{2i}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}}\Big{|}\Big{\langle}f\oplus
g,g_{1i}\oplus g_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle=\sum_{i\in\sigma}\Big{|}\Big{\langle}f,f_{1i}\Big{\rangle}+\Big{\langle}g,f_{2i}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}}\Big{|}\Big{\langle}f,g_{1i}\Big{\rangle}+\Big{\langle}g,g_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle=\sum_{i\in\sigma\cap\mathbb{Z}^{-}}\Big{|}\Big{\langle}f,f_{1i}\Big{\rangle}+\Big{\langle}g,f_{2i}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma\cap\mathbb{Z}^{+}}\Big{|}\Big{\langle}f,f_{1i}\Big{\rangle}+\Big{\langle}g,f_{2i}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}\cap\mathbb{Z}^{-}}\Big{|}\Big{\langle}f,g_{1i}\Big{\rangle}+\Big{\langle}g,g_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle\quad\quad\quad\quad+\sum_{i\in\sigma^{c}\cap\mathbb{Z}^{+}}\Big{|}\Big{\langle}f,g_{1i}\Big{\rangle}+\Big{\langle}g,g_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle=\sum_{i\in\sigma\cap\mathbb{Z}^{-}}\Big{|}\Big{\langle}f,f_{1i}\Big{\rangle}+\Big{\langle}g,0\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma\cap\mathbb{Z}^{+}}\Big{|}\Big{\langle}f,0\Big{\rangle}+\Big{\langle}g,f_{2i}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}\cap\mathbb{Z}^{-}}\Big{|}\Big{\langle}f,g_{1i}\Big{\rangle}+\Big{\langle}g,0\Big{\rangle}\Big{|}^{2}$
$\displaystyle\quad\quad\quad\quad+\sum_{i\in\sigma^{c}\cap\mathbb{Z}^{+}}\Big{|}\Big{\langle}f,0\Big{\rangle}+\Big{\langle}g,g_{2i}\Big{\rangle}\Big{|}^{2}$
$\displaystyle=\sum_{i\in\sigma\cap\mathbb{Z}^{-}}\Big{|}\Big{\langle}f,\phi_{\Theta^{1}(i)}^{1}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma\cap\mathbb{Z}^{+}}\Big{|}\Big{\langle}g,\phi_{\Theta^{2}(i)}^{2}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}\cap\mathbb{Z}^{-}}\Big{|}\Big{\langle}f,\psi_{\Theta^{1}(i)}^{1}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}\cap\mathbb{Z}^{+}}\Big{|}\Big{\langle}g,\psi_{\Theta^{2}(i)}^{2}\Big{\rangle}\Big{|}^{2}$
$\displaystyle=\left(\sum_{i\in\sigma\cap\mathbb{Z}^{-}}\Big{|}\Big{\langle}f,\phi_{\Theta^{1}(i)}^{1}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}\cap\mathbb{Z}^{-}}\Big{|}\Big{\langle}f,\psi_{\Theta^{1}(i)}^{1}\Big{\rangle}\Big{|}^{2}\right)+\left(\sum_{i\in\sigma\cap\mathbb{Z}^{+}}\Big{|}\Big{\langle}g,\phi_{\Theta^{2}(i)}^{2}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}\cap\mathbb{Z}^{+}}\Big{|}\Big{\langle}g,\psi_{\Theta^{2}(i)}^{2}\Big{\rangle}\Big{|}^{2}\right)$
$\displaystyle=\left(\sum_{i\in\Theta^{1}(\sigma\cap\mathbb{Z}^{-})}\Big{|}\Big{\langle}f,\phi_{i}^{1}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\Theta^{1}(\sigma^{c}\cap\mathbb{Z}^{-})}\Big{|}\Big{\langle}f,\psi_{i}^{1}\Big{\rangle}\Big{|}^{2}\right)$
$\displaystyle\quad\quad\quad\quad+\left(\sum_{i\in\Theta^{2}(\sigma\cap\mathbb{Z}^{+})}\Big{|}\Big{\langle}g,\phi_{i}^{2}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\Theta^{2}(\sigma^{c}\cap\mathbb{Z}^{+})}\Big{|}\Big{\langle}g,\psi_{i}^{2}\Big{\rangle}\Big{|}^{2}\right).$
(3.1)

Let
$\sigma_{1}=\Theta^{1}(\sigma\cap\mathbb{Z}^{-}),\sigma_{2}=\Theta^{2}(\sigma\cap\mathbb{Z}^{+})$.
Then, $\sigma_{1}$ is a subset of $I_{1}$ such that the compliment of
$\sigma_{1}$ in $I_{1}$ is $\Theta^{1}(\sigma^{c}\cap\mathbb{Z}^{-})$ and
$\sigma_{2}$ is a subset of $I_{2}$ such that the compliment of $\sigma_{2}$
in $I_{2}$ is $\Theta^{2}(\sigma^{c}\cap\mathbb{Z}^{+})$. By using the fact
that $\Phi_{i}$ and $\Psi_{i}$ are woven frames for the Hilbert space
$\mathcal{H}_{i}\ (i=1,2)$ with universal bounds $A_{i},B_{i}$ (say) and using
(3), we compute

$\displaystyle\min\\{A_{1},A_{2}\\}(\|f\|^{2}+\|g\|^{2})$ $\displaystyle\leq
A_{1}\|f\|^{2}+A_{2}\|g\|^{2}$
$\displaystyle\leq\left(\sum_{i\in\sigma_{1}}\Big{|}\Big{\langle}f,\phi_{i}^{1}\Big{\rangle}\Big{|}^{2}+\sum_{i\in
I_{1}\setminus\sigma_{1}}\Big{|}\Big{\langle}f,\psi_{i}^{1}\Big{\rangle}\Big{|}^{2}\right)+\left(\sum_{i\in\sigma_{2}}\Big{|}\Big{\langle}g,\phi_{i}^{2}\Big{\rangle}\Big{|}^{2}+\sum_{i\in
I_{2}\setminus\sigma_{2}}\Big{|}\Big{\langle}g,\psi_{i}^{2}\Big{\rangle}\Big{|}^{2}\right)$
$\displaystyle=\sum_{i\in\sigma}\Big{|}\Big{\langle}f\oplus g,f_{1i}\oplus
f_{2i}\Big{\rangle}\Big{|}^{2}+\sum_{i\in\sigma^{c}}\Big{|}\Big{\langle}f\oplus
g,g_{1i}\oplus g_{2i}\Big{\rangle}\Big{|}^{2}$ $\displaystyle\leq
B_{1}\|f\|^{2}+B_{2}\|g\|^{2}$
$\displaystyle\leq\max\\{B_{1},B_{2}\\}(\|f\|^{2}+\|g\|^{2}).$

Hence the vector-valued frames $\\{F_{1},F_{2},\mathbb{Z}^{*}\\}$ and
$\\{G_{1},G_{2},\mathbb{Z}^{*}\\}$ are woven with universal frame bounds
$\min\\{A_{1},A_{2}\\}$ and $\max\\{B_{1},B_{2}\\}$. ∎

##  4\. Vector-valued Weaving Riesz Bases

######  Definition 4.1.

The family $\\{F_{1},\dots,F_{L},I\\}$ $(F_{i}=\\{f_{ik}\\}_{k\in
I}\subseteq\mathcal{H}_{i}\ (1\leq i\leq L))$ is called a _vector-valued Riesz
basis_ (or _super-Riesz basis_) for the space
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ if there exists a bounded
bijective operator
$U:\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}\rightarrow\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$
such that $U(e_{1k}\oplus\cdots\oplus e_{Lk})=f_{1k}\oplus\cdots\oplus f_{Lk}$
$(k\in I)$, where $\\{e_{1k}\oplus\cdots\oplus e_{Lk}\\}_{k\in I}$ is an
orthonormal basis for $\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$.

A characterization of vector-valued Riesz basis for a Hilbert space
$\mathcal{H}_{1}\oplus....\oplus\mathcal{H}_{L}$ can be obtained by using [7,
Theorem 3.6.6]

######  Theorem 4.2.

For a sequence $\\{F_{1},\dots,F_{L},I\\}$ $(F_{i}=\\{f_{ik}\\}_{k\in
I}\subseteq\mathcal{H}_{i})$, the following conditions are equivalent:

  1. $(i)$

$\\{F_{1},\dots,F_{L},I\\}$ is a vector-valued Riesz basis for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$.

  2. $(ii)$

$\\{F_{1},\dots,F_{L},I\\}$ is complete in
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ and there exist finite
constants $A,B>0$ such that for every finite scalar sequence $\\{c_{k}\\}$,
one has

$\displaystyle A\sum|c_{k}|^{2}\leq\Big{\|}\sum c_{k}(f_{1k}\oplus\cdots\oplus
f_{Lk})\Big{\|}^{2}\leq B\sum|c_{k}|^{2}.$ (4.1)

The scalars $A$ and $B$ are called lower and upper _Riesz bounds_ ,
respectively.

######  Definition 4.3.

The family $\\{F_{1},\dots,F_{L},I\\}$ $(F_{i}=\\{f_{ik}\\}_{k\in
I}\subseteq\mathcal{H}_{i})$ is called a _vector-valued Riesz sequence_ (or
_super-Riesz sequence_) in the space
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ if (4.1) holds for all
finite scalar sequences $\\{c_{k}\\}_{k\in I}$.

The following proposition provides a sufficient condition for vector-valued
Riesz basis.

######  Proposition 4.4.

Suppose $\\{F_{1},\dots,F_{L},I\\}$ $(\text{where}\ F_{i}=\\{f_{ik}\\}_{k\in
I})$ is a vector-valued frame for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ such that for some
$n\in[L]$, the sequence $F_{n}$ is a Riesz basis for $\mathcal{H}_{n}$. Then,
the family $\\{F_{1},\dots,F_{L},I\\}$ is a vector-valued Riesz basis for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$.

###### Proof.

Suppose that $F_{n}$ is a Riesz basis for $\mathcal{H}_{n}$ with lower and
upper Riesz bounds $A_{n}$ and $B_{n}$, respectively. Then, for any finite
scalar sequence $\\{c_{k}\\}_{k\in I}$, we compute

$\displaystyle\Big{\|}\sum_{k\in I}c_{k}\Big{(}f_{1k}\oplus\cdots\oplus
f_{Lk}\Big{)}\Big{\|}^{2}$ $\displaystyle=\Big{\|}\sum_{k\in
I}c_{k}f_{1k}\Big{\|}^{2}+\cdots+\Big{\|}\sum_{k\in I}c_{k}f_{Lk}\Big{\|}^{2}$
$\displaystyle\geq\Big{\|}\sum_{k\in I}c_{k}f_{nk}\Big{\|}^{2}$
$\displaystyle\geq A_{n}\sum_{k\in I}|c_{k}|^{2}.$

Therefore, (4.1) is satisfied. By hypothesis, the family
$\\{F_{1},\dots,F_{L},I\\}$ is a vector-valued frame for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$. Thus, the family
$\\{F_{1},\dots,F_{L},I\\}$ is complete in
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$. Hence
$\\{F_{1},\dots,F_{L},I\\}$ is a vector-valued Riesz basis for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$. ∎

######  Remark 4.5.

If a family $\\{F_{1},\dots,F_{L},I\\}$ is a vector-valued Riesz basis for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$, then its component $F_{k}$
need not be a Riesz basis for the atomic space $\mathcal{H}_{k}$. Indeed, let
$L=2,I=\mathbb{N}$ and $\mathcal{H}_{1}=\mathcal{H}_{2}=L^{2}(I,\mu)$, where
$\mu$ is the counting measure. Define $F_{1}=\\{f_{1i}\\}_{i\in
I},F_{2}=\\{f_{2i}\\}_{i\in I}$ as follows :

$\displaystyle f_{1i}=$ $\displaystyle\begin{cases}e_{j},\ i=2j-1\
(j\in\mathbb{N})\\\ 0,\ i=2j\ (j\in\mathbb{N}).\end{cases};\quad\quad
f_{2i}=\begin{cases}0,\ i=2j-1\ (j\in\mathbb{N})\\\ e_{j},\ i=2j\
(j\in\mathbb{N}).\end{cases}$

where $\\{e_{i}\\}_{i\in I}$ is the canonical orthonormal basis of
$\mathcal{H}_{1}$.
Then, $\\{F_{1},F_{2},I\\}$ is a vector-valued Riesz basis for
$\mathcal{H}_{1}\oplus\mathcal{H}_{2}$. But $F_{1}$ and $F_{2}$ are not Riesz
bases for $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, respectively.

######  Definition 4.6.

A vector-valued family
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$
$(F_{j}^{i}=\\{f_{jk}^{i}\\}_{k\in I})$ for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ is said to be a _woven
Riesz sequence_ if there exist universal constants $A$ and $B$ such that for
any partition $\\{\sigma_{i}\\}_{i\in[m]}$ of $I$, one has

$\displaystyle
A\sum|c_{k}|^{2}\leq\Big{\|}\sum_{i\in[m]}\sum_{k\in\sigma_{i}}c_{k}(f_{1k}^{i}\oplus\cdots\oplus
f_{Lk}^{i})\Big{\|}^{2}\leq B\sum|c_{k}|^{2}$

for all finite scalar sequences $\\{c_{k}\\}_{k\in I}$.

Let us have a look at Example 3.6 which shows that if the family of frames for
atomic spaces are woven, then the family of vector-valued frames need not be
woven. But this is not the case for woven Riesz sequences. More precisely, if
a family of Bessel sequences for atomic spaces is a woven Riesz sequence, then
its associated family of vector-valued Bessel sequences is a woven Riesz
sequence.

######  Theorem 4.7.

Suppose $\\{F_{1}^{i},\dots,F_{L}^{i},I\\}$ is a vector-valued family for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ such that
$F_{j}^{i}=\\{f_{jk}^{i}\\}_{k\in I}$ is a Bessel sequence for all $i\in[m]$
and $j\in[L]$. If for some $n\in[L]$, the family
$\Big{\\{}F_{n}^{i}:i\in[m]\Big{\\}}$ is a woven Riesz sequence, then
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$ is a woven Riesz
sequence.

###### Proof.

Let $A$ and $B$ be universal Riesz bounds for the family
$\Big{\\{}F_{n}^{i}:i\in[m]\Big{\\}}$. Let $\\{\sigma_{i}\\}_{i\in[m]}$ be any
partition of $I$. Then, for any $\\{c_{k}\\}_{k\in I}\in\ell^{2}(I)$, we
compute

$\displaystyle\Big{\|}\sum_{i\in[m]}\sum_{k\in\sigma_{i}}c_{k}\Big{(}f_{1k}^{i}\oplus\cdots\oplus
f_{Lk}^{i}\Big{)}\Big{\|}^{2}$
$\displaystyle=\Big{\|}\sum_{i\in[m]}\sum_{k\in\sigma_{i}}c_{k}f_{1k}^{i}\Big{\|}^{2}+\cdots+\Big{\|}\sum_{i\in[m]}\sum_{k\in\sigma_{i}}c_{k}f_{Lk}^{i}\Big{\|}^{2}$
$\displaystyle\geq\Big{\|}\sum_{i\in[m]}\sum_{k\in\sigma_{i}}c_{k}f_{nk}^{i}\Big{\|}^{2}$
$\displaystyle\geq A\sum_{k\in I}|c_{k}|^{2}.$

By Corollary 3.8,
$\bigcup\limits_{i\in[m]}\\{F_{1}^{i},\dots,F_{L}^{i},\sigma_{i}\\}$ is a
Bessel sequence with universal bound $B_{o}$ (say). Therefore,

$\displaystyle\Big{\|}\sum_{i\in[m]}\sum_{k\in\sigma_{i}}c_{k}\Big{(}f_{1k}^{i}\oplus\cdots\oplus
f_{Lk}^{i}\Big{)}\Big{\|}^{2}\leq B_{o}\sum_{k\in I}|c_{k}|^{2}.$

Hence the family $\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$
is a woven Riesz sequence with universal bounds $A$ and $B_{o}$. ∎

A vector-valued frame with one component as Riesz basis (for the underlying
atomic space) cannot be weave with vector-valued frames whose corresponding
component is not a Riesz basis.

######  Proposition 4.8.

Suppose the family of vector-valued frames
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$ for
$\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ is woven and $F_{p}^{q}$ is
a Riesz basis for $\mathcal{H}_{p}$ for some $p\in[L]$ and $q\in[m]$. Then,
$F_{p}^{i}$ is a Riesz basis for $\mathcal{H}_{p}$ for all $i\in[m]$. In
particular, $\\{F_{1}^{i},\dots,F_{L}^{i},I\\}$ is a vector-valued Riesz basis
for $\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ for all $i\in[m]$.

###### Proof.

Since family of vector-valued frames
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$ is woven, the
family $\Big{\\{}F_{j}^{i}:i\in[m]\Big{\\}}$ is woven ($j\in[L]$). In
particular, $\Big{\\{}F_{p}^{i}:i\in[m]\Big{\\}}$ is woven. By hypothesis,
$F_{p}^{q}$ is a Riesz basis for $\mathcal{H}_{p}$. Therefore, by Theorem 1.4,
$F_{p}^{i}$ is a Riesz basis for $\mathcal{H}_{p}$ for all $i\in[m]$. Hence by
Proposition 4.4, the family $\\{F_{1}^{i},\dots,F_{L}^{i},I\\}$ is a vector-
valued Riesz basis for $\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$ for
all $i\in[m]$. ∎

The following theorem provide sufficient conditions for vector-valued weaving
Riesz sequences in terms of operators on atomic spaces.

######  Theorem 4.9.

Suppose $\\{F_{1}^{i},\dots,F_{L}^{i},\mathbb{N}\\}$ $(i\in[m])$ is a vector-
valued family for $\mathcal{H}_{1}\oplus\cdots\oplus\mathcal{H}_{L}$. Let
there exists some $n\in[L]$ such that
$F_{j}^{i}=\\{f_{jk}^{i}\\}_{k\in\mathbb{N}}$ is a Bessel sequence for
$j\in[L]\setminus\\{n\\}$ $(i\in[m])$. Let $\\{\chi_{k}\\}_{k\in\mathbb{N}}$
and $\\{e_{k}\\}_{k\in\mathbb{N}}$ be any orthonormal bases for
$\mathcal{H}_{n}$ such that

$\displaystyle
f_{nk}^{i}=\chi_{k}-\sum\limits_{p\in\mathbb{N}}\alpha_{kp}^{i}T_{p}^{i}(e_{k})\
\text{for all}\ i\in[m]\ \text{and for all}\ k\in\mathbb{N},$ and
$\displaystyle\lambda_{i}=\sum\limits_{p\in\mathbb{N}}\|T_{p}^{i}\|\sup\limits_{k\in\mathbb{N}}|\alpha_{kp}^{i}|<\frac{1}{3^{\frac{m-1}{2}}}\
\ \text{for all}\ i\in[m],$

where $T_{p}^{i}\in\mathcal{B}(\mathcal{H}_{n})$ for each $p\in\mathbb{N},\
i\in[m]$ and $\alpha_{kp}^{i}$ are scalars for all $k,\ p\in\mathbb{N}$,
$i\in[m]$ . Then,
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},I\\}:i\in[m]\Big{\\}}$ is a woven Riesz
sequence.

###### Proof.

Let $\\{\sigma_{i}\\}_{i\in[m]}$ be any partition of $\mathbb{N}$. Then, for
any $N\in\mathbb{N}$ and scalars $c_{1},c_{2},\dots,c_{N}$, we compute

$\displaystyle\Big{\|}\sum_{i\in[m]}\sum_{k\in[N]\bigcap\sigma_{i}}c_{k}(\chi_{k}-f_{nk}^{i})\Big{\|}$
$\displaystyle=\Big{\|}\sum_{i\in[m]}\sum_{k\in[N]\bigcap\sigma_{i}}c_{k}\Big{(}\sum\limits_{p\in\mathbb{N}}\alpha_{kp}^{i}T_{p}^{i}(e_{k})\Big{)}\Big{\|}$
$\displaystyle=\Big{\|}\sum\limits_{p\in\mathbb{N}}\sum_{i\in[m]}\sum_{k\in[N]\bigcap\sigma_{i}}c_{k}\alpha_{kp}^{i}T_{p}^{i}(e_{k})\Big{\|}$
$\displaystyle\leq\sum\limits_{p\in\mathbb{N}}\Big{\|}\sum_{i\in[m]}\sum_{k\in[N]\bigcap\sigma_{i}}c_{k}\alpha_{kp}^{i}T_{p}^{i}(e_{k})\Big{\|}$
$\displaystyle\leq\sum\limits_{p\in\mathbb{N}}\sum_{i\in[m]}\Big{\|}T_{p}^{i}\Big{(}\sum_{k\in[N]\bigcap\sigma_{i}}c_{k}\alpha_{kp}^{i}e_{k}\Big{)}\Big{\|}$
$\displaystyle\leq\sum\limits_{p\in\mathbb{N}}\sum_{i\in[m]}\|T_{p}^{i}\|\Big{\|}\sum_{k\in[N]\bigcap\sigma_{i}}c_{k}\alpha_{kp}^{i}e_{k}\Big{\|}$
$\displaystyle=\sum\limits_{p\in\mathbb{N}}\sum_{i\in[m]}\|T_{p}^{i}\|\Big{(}\sum_{k\in[N]\bigcap\sigma_{i}}|c_{k}\alpha_{kp}^{i}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\sum\limits_{p\in\mathbb{N}}\sum_{i\in[m]}\|T_{p}^{i}\|\sup\limits_{k\in\mathbb{N}}|\alpha_{kp}^{i}|\Big{(}\sum_{k\in[N]\bigcap\sigma_{i}}|c_{k}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle=\sum_{i\in[m]}\lambda_{i}\Big{(}\sum_{k\in[N]\bigcap\sigma_{i}}|c_{k}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\max\limits_{i\in[m]}\\{\lambda_{i}\\}\sum_{i\in[m]}\Big{(}\sum_{k\in[N]\bigcap\sigma_{i}}|c_{k}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\max\limits_{i\in[m]}\\{\lambda_{i}\\}3^{\frac{m-1}{2}}\Big{(}\sum_{i\in[m]}\sum_{k\in[N]\bigcap\sigma_{i}}|c_{k}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle=\lambda\Big{(}\sum_{k\in[N]}|c_{k}|^{2}\Big{)}^{\frac{1}{2}},\text{
where }\lambda=3^{\frac{m-1}{2}}\max\limits_{i\in[m]}\\{\lambda_{i}\\}<1.$

By Theorem 2.1, $\bigcup\limits_{i\in[m]}\\{f_{nk}^{i}\\}_{k\in\sigma_{i}}$ is
equivalent to orthonormal basis $\\{\chi_{k}\\}_{k\in\mathbb{N}}$. Therefore,
there exists a bijective linear operator
$U:\mathcal{H}_{n}\rightarrow\mathcal{H}_{n}$ such that
$U(\chi_{k})=f_{nk}^{i}$ for $k\in\sigma_{i}$. Thus,
$\bigcup\limits_{i\in[m]}\\{f_{nk}^{i}\\}_{k\in\sigma_{i}}$ is a Riesz basis
and hence a frame for $\mathcal{H}_{n}$. Therefore, the family of frames
$\Big{\\{}\\{f_{nk}^{i}\\}_{k\in\mathbb{N}}:i\in[m]\Big{\\}}$ is weakly woven
and hence woven (by Remark 1.3). Thus, there exist universal frame bounds
(note that the Riesz basis bounds coincide with the frame bounds) for the
family $\Big{\\{}\\{f_{nk}^{i}\\}_{k\in\mathbb{N}}:i\in[m]\Big{\\}}$. That is,
$\Big{\\{}\\{f_{nk}^{i}\\}_{k\in\mathbb{N}}:i\in[m]\Big{\\}}$ is woven Riesz
sequence. Hence by Theorem 4.7, the family
$\Big{\\{}\\{F_{1}^{i},\dots,F_{L}^{i},\mathbb{N}\\}:i\in[m]\Big{\\}}$ is a
woven Riesz sequence. ∎

To conclude the paper, we illustrates Theorem 4.9 with the following example.

######  Example 4.10.

Let $L=m=n=2$ and let $\mathcal{H}_{1}=\mathcal{H}_{2}=\ell^{2}(\mathbb{N})$.
Suppose that $\\{\chi_{k}\\}_{k\in\mathbb{N}}$ and
$\\{e_{k}\\}_{k\in\mathbb{N}}$ are any two orthonormal basis for
$\ell^{2}(\mathbb{N})$.

Choose $0<\epsilon<\frac{1}{\sqrt{3}\sum\limits_{k\in\mathbb{N}}1/k^{2}}$.
Define sequences
$\\{T_{p}^{i}\\}_{p\in\mathbb{N}}\subset\mathcal{B}(\mathcal{H}_{2})$
$(i=1,2)$ by

$\displaystyle T_{p}^{1}:\mathcal{H}_{2}\rightarrow\mathcal{H}_{2}\text{ by
}T_{p}^{1}(\\{\xi_{1},\xi_{2},\dots\\})=\\{0,0,\dots,0,\xi_{p},\xi_{p+1},\dots\\},$
$\displaystyle T_{p}^{2}:\mathcal{H}_{2}\rightarrow\mathcal{H}_{2}\text{ by
}T_{p}^{2}(\\{\xi_{1},\xi_{2},\dots\\})=\Big{\\{}\frac{\xi_{p}}{4},\frac{\xi_{p+1}}{4},\dots\Big{\\}}.$

Choose $\\{\alpha_{kp}^{i}\\}_{k\in\mathbb{N}}\subset\mathbb{K}\
(p\in\mathbb{N};\ i=1,2)$ as follows:

$\displaystyle\\{\alpha_{kp}^{1}\\}_{k\in\mathbb{N}}=\Big{\\{}0,\dots,0,\underbrace{\frac{\epsilon}{p^{2}}}_{p^{th}-place},\frac{\epsilon}{(p+1)^{2}},\dots\Big{\\}}\
(p\in\mathbb{N}),$
$\displaystyle\\{\alpha_{kp}^{2}\\}_{k\in\mathbb{N}}=\Big{\\{}0,\dots,0,\underbrace{\frac{4\epsilon}{(p+1)^{2}}}_{p+1^{th}-place},\frac{4\epsilon}{(p+2)^{2}},\dots\Big{\\}},\
p\in\mathbb{N}.$

Then

$\displaystyle\lambda_{1}=\sum\limits_{p\in\mathbb{N}}\|T_{p}^{1}\|\sup\limits_{k\in\mathbb{N}}|\alpha_{kp}^{i}|\leq\sum\limits_{p\in\mathbb{N}}\frac{\epsilon}{p^{2}}<\frac{1}{\sqrt{3}},$
and
$\displaystyle\lambda_{2}=\sum\limits_{p\in\mathbb{N}}\|T_{p}^{2}\|\sup\limits_{k\in\mathbb{N}}|\alpha_{kp}^{i}|\leq\sum\limits_{p\in\mathbb{N}}\frac{1}{4}\frac{4\epsilon}{(p+1)^{2}}\leq\sum\limits_{p\in\mathbb{N}}\frac{\epsilon}{p^{2}}<\frac{1}{\sqrt{3}}.$

Define $\\{F_{1}^{1},F_{2}^{1},\mathbb{N}\\}$ and
$\\{F_{1}^{2},F_{2}^{2},\mathbb{N}\\}$ as follows:

$\displaystyle
f_{1k}^{1}=\Big{\\{}0,\dots,0,\underbrace{1}_{k^{th}-place},1,0,0,\dots\Big{\\}},\
k\in\mathbb{N},$ $\displaystyle
f_{2k}^{1}=\chi_{k}-\sum\limits_{p\in\mathbb{N}}\alpha_{kp}^{1}T_{p}^{1}(e_{k}),\
k\in\mathbb{N},$ $\displaystyle
f_{1k}^{2}=\Big{\\{}0,\dots,0,\underbrace{1}_{k^{th}-place},1,1,0,0,\dots\Big{\\}},\
k\in\mathbb{N},$ $\displaystyle
f_{2k}^{2}=\chi_{k}-\sum\limits_{p\in\mathbb{N}}\alpha_{kp}^{2}T_{p}^{2}(e_{k}),\
k\in\mathbb{N}.$

It is easy to verify that $F_{j}^{i}$ is a Bessel sequence for
$\mathcal{H}_{i}$, for all $i\in[m],\ j\in[L]\setminus\\{n\\}=\\{1\\}$. Hence,
by Theorem 4.9, the family
$\Big{\\{}\\{F_{1}^{i},F_{2}^{i},\mathbb{N}\\}:i\in[2]\Big{\\}}$ is a woven
Riesz sequence.

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