# Weaving $K$-Frames in Hilbert Spaces

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

Gǎvruta introduced $K$-frames for Hilbert spaces to study atomic systems with
respect to a bounded linear operator. There are many differences between
$K$-frames and standard frames, so we study weaving properties of $K$-frames.
Two frames $\\{\phi_{i}\\}_{i\in I}$ and $\\{\psi_{i}\\}_{i\in I}$ for a
separable Hilbert space $\mathcal{H}$ 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 $\mathcal{H}$ with frame bounds $A,B$. In this paper, we present necessary
and sufficient conditions for weaving $K$-frames in Hilbert spaces. It is
shown that woven $K$-frames and weakly woven $K$-frames are equivalent.
Finally, sufficient conditions for Paley-Wiener type perturbation of weaving
$K$-frames are given.

###### Key words and phrases:

Frames; $K$-frames; weaving frames; local atoms; perturbation.
The research of Deepshikha is supported by the Council of Scientific $\&$
Industrial Research (CSIR) (Grant No.: 09/045(1352)/2014- EMR-I), India. Lalit
was supported by R $\&$ D Doctoral Research Programme, University of Delhi
(Grant No.: RC/2015/9677).
∗Corresponding author-Lalit K. Vashisht

###### 2010 Mathematics Subject Classification:

42C15; 42C30; 42C40.

##  1\. Introduction and Preliminaries

Let $\mathcal{H}$ be a complex separable Hilbert space with an inner product
$\langle.,.\rangle$. A countable sequence
$\\{f_{k}\\}_{k=1}^{\infty}\subset\mathcal{H}$ is a _frame_ for $\mathcal{H}$
if there exist positive scalars $A_{o}\leq B_{o}<\infty$ such that

(1.1) $\displaystyle A_{o}\|f\|^{2}\leq\sum_{k=1}^{\infty}|\langle
f,f_{k}\rangle|^{2}\leq B_{o}\|f\|^{2}\ \text{for all}\ f\in\mathcal{H}.$

The scalars $A_{o}$ and $B_{o}$ are called _lower_ and _upper frame bounds_ ,
respectively. The sequence $\\{f_{k}\\}_{k=1}^{\infty}$ is called a _Bessel
sequence_ with _Bessel bound_ $B_{o}$ if the upper inequality in (1.1) holds
for all $f\in\mathcal{H}$.

Following three operators are associated with a frame
$\\{f_{k}\\}_{k=1}^{\infty}$ for $\mathcal{H}$:

$\displaystyle\text{\emph{pre-frame operator}}\ \
T:\ell^{2}(\mathbb{N})\rightarrow\mathcal{H},\quad\
T\\{c_{k}\\}_{k=1}^{\infty}=\sum\limits_{k=1}^{\infty}c_{k}f_{k},\
\\{c_{k}\\}_{k=1}^{\infty}\in\ell^{2}(\mathbb{N}),$
$\displaystyle\text{\emph{analysis operator \ $($adjoint of $T)$}}\ \
T^{*}:\mathcal{H}\rightarrow\ell^{2}(\mathbb{N}),\quad T^{*}f=\\{\langle
f,f_{k}\rangle\\}_{k=1}^{\infty},f\in\mathcal{H},$
$\displaystyle\text{\emph{frame operator}}\quad
S=TT^{*}:\mathcal{H}\rightarrow\mathcal{H},\quad
Sf=\sum\limits_{k=1}^{\infty}\langle f,f_{k}\rangle f_{k},f\in\mathcal{H}.$

The frame operator $S$ is a bounded, linear and invertible operator on
$\mathcal{H}$. This gives the _reconstruction_ of each vector
$f\in\mathcal{H}$,

$\displaystyle f=SS^{-1}f=\sum\limits_{k=1}^{\infty}\langle
S^{-1}f,f_{k}\rangle f_{k}.$

Thus, a frame for $\mathcal{H}$ allows each vector in $\mathcal{H}$ to be
written as a linear combination of the elements in the frame, but the linear
independence between the frame elements is not required. Concerning the
evolution of the notion of Hilbert frames and their applications in different
directions in science and engineering, it is necessary to mention the nobel
books by Casazza and Kutyniok [4], Christensen [7], Gröchenig [19]; and
beautiful research tutorials by Casazza [5] and Casazza and Lynch [6].

Next we give some basic notations. The family of all bounded linear operators
from a Banach space $\mathcal{X}$ into a Banach space $\mathcal{Y}$ is denoted
by $\mathcal{B}(\mathcal{X},\mathcal{Y})$. If $\mathcal{X}=\mathcal{Y}$, then
we write $\mathcal{B}(\mathcal{X},\mathcal{Y})=\mathcal{B}(\mathcal{X})$. The
range and the kernel of $K\in\mathcal{B}(\mathcal{X},\mathcal{Y})$ are denoted
by $R(K)$ and $N(K)$, respectively. The pseudo-inverse of
$K\in\mathcal{B}(\mathcal{H})$ is denoted by $K^{\dagger}$. Note that
$KK^{\dagger}f=f$ for all $f\in R(K)$. Throughout the paper $R(K)$ is closed.
By $\mathbb{N}$ we denote the set of all positive integers. The canonical
orthonormal basis for $\ell^{2}(\mathbb{N})$ is the sequence
$\\{e_{n}\\}_{n=1}^{\infty}$, where $e_{n}=\\{0,0,0,\dots,\underbrace{1}_{nth\
place},0,0,0,\dots\\}$ for all $n\in\mathbb{N}$. For a sequence of vectors
$\\{f_{k}\\}_{k\in I}\subset\mathcal{H}$, the closure of the
span$\\{f_{k}\\}_{k\in I}$ is denoted by $[f_{k}]_{k\in I}$.

The following key-theorem can be found in [13]

######  Theorem 1.1.

[13] Let $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ be separable Hilbert spaces
and let $L_{1}\in\mathcal{B}(\mathcal{H}_{1},\mathcal{H})$,
$L_{2}\in\mathcal{B}(\mathcal{H}_{2},\mathcal{H})$. The following statements
are equivalent:

  1. $(i)$

$R(L_{1})\subset R(L_{2})$

  2. $(ii)$

$L_{1}L_{1}^{*}\leq\lambda^{2}L_{2}L_{2}^{*}$ for some $\lambda\geq 0$ and

  3. $(iii)$

there exists a $C\in\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})$ such that
$L_{1}=L_{2}C$ and $\|C\|^{2}=\inf\\{\mu:L_{1}L_{1}^{*}\leq\mu
L_{2}L_{2}^{*}\\}$.

###  1.1. $K$-frames in Hilbert spaces

Feichtinger and Werther [14] introduced a family of analysis and synthesis
systems with frame-like properties for closed subspaces of $\mathcal{H}$ and
call it an _atomic system_ (or _local atoms_). The motivation for the atomic
system is based on examples arising in sampling theory, see [15]. One of the
important properties of the atomic system is that it can generate a proper
subspace even though they do not belong to them.

######  Definition 1.2.

[14] Let $\mathcal{H}_{o}$ be a closed subspace of $\mathcal{H}$. A sequence
$\\{f_{k}\\}_{k=1}^{\infty}\subset\mathcal{H}$ is called a _family of local
atoms_ (or _atomic system_) for $\mathcal{H}_{o}$, if

  1. $(i)$

$\\{f_{k}\\}_{k=1}^{\infty}$ is Bessel sequence in $\mathcal{H}$,

  2. $(ii)$

there exists a sequence of linear functionals $\\{c_{k}\\}_{k=1}^{\infty}$ and
a real number $C>0$ such that

$\displaystyle\sum_{k=1}^{\infty}|c_{k}(f)|^{2}\leq C\|f\|^{2}\ \text{for
all}\ f\in\mathcal{H}_{o},$

  3. $(iii)$

$f=\sum_{k=1}^{\infty}c_{k}(f)f_{k}\ \text{for all}\ f\in\mathcal{H}_{o}.$

######  Remark 1.3.

Note that the linear functionals $\\{c_{k}\\}_{k=1}^{\infty}$ need only to be
defined on the subspace $\mathcal{H}_{o}$. We say that $c_{k}$ are _associated
functionals_ of the local atoms $\\{f_{k}\\}$. The constant $C$ is called the
_atomic bound_. Furthermore, the partial sum $\sum_{k=1}^{N}c_{k}(f)f_{k}$ of
the series in (iii) can be converges to $f$ from “outside” of
$\mathcal{H}_{o}$. The family of local atoms for $\mathcal{H}_{o}$ is a so-
called _pseudo-frame_ as proposed by Li and Ogawa in [20].

Gǎvruta in [16] introduced and studied $K$-frames in Hilbert spaces to study
atomic systems with respect to a bounded linear operator $K$ on Hilbert
spaces.

######  Definition 1.4.

[16] Let $K\in\mathcal{B}(\mathcal{H})$. A sequence
$\\{f_{k}\\}_{k=1}^{\infty}\subset\mathcal{H}$ is called a _$K$ -frame_ for
$\mathcal{H}$, if there exist constants $A,B>0$ such that

(1.2) $\displaystyle A\|K^{*}f\|^{2}\leq\sum_{k=1}^{\infty}|\langle
f,f_{k}\rangle|^{2}\leq B\|f\|^{2}\ \text{for all}\ f\in\mathcal{H}.$

The numbers $A$ and $B$ are called _lower_ and _upper_ _$K$ -frame bounds_,
respectively. If $I$ is the identity operator on $\mathcal{H}$, then
$K$-frames are the standard frames. $K$-frames are more general than standard
frames in the sense that the lower frame bound only holds for the elements in
the range of $K^{*}$. Since a $K$-frame $\\{f_{k}\\}_{k=1}^{\infty}$ for
$\mathcal{H}$ is a Bessel sequence, we can define the pre-frame operator,
analysis operator and frame operator associated with
$\\{f_{k}\\}_{k=1}^{\infty}$. The frame operator of a $K$-frame is not
invertible on $\mathcal{H}$ in general, but it is invertible on a subspace
$R(K)$, where the range $R(K)\subset\mathcal{H}$ is closed. Furthermore, there
are many differences between $K$-frames and standard frames, see [16]. Gǎvruta
[16] characterize $K$-frames in Hilbert spaces by using bounded linear
operators. In [8, 17, 18, 24] some new results about $K$-frames were obtained.

###  1.2. Background on weaving frames

The concept of _“weaving frames”_ for complex separable Hilbert spaces
introduced by Bemrose, Casazza, Gröchenig, Lammers and Lynch in [1]. For a
fixed $m\in\mathbb{N}$, we write $[m]=\\{1,2,\dots,m\\}\quad\text{ and }\
[m]^{c}=\mathbb{N}\setminus[m]=\\{m+1,m+2,\dots\\}$.

######  Definition 1.5.

[1] Let $I$ be a countable indexing set. A family of frames
$\Big{\\{}\\{\phi_{ij}\\}_{j\in I}:i\in[m]\Big{\\}}$ for $\mathcal{H}$ is said
to be _woven_ , if there are universal constants $A$ and $B$ such that for
every partition $\\{\sigma_{i}\\}_{i\in[m]}$ of $I$, the family
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}}$ is a frame for
$\mathcal{H}$ with frame bounds $A$ and $B$.

This new notion of weaving frames is motivated by a problem in distributed
signal processing. Weaving frames has potential applications in wireless
sensor networks that require distributed processing under different frames, as
well as pre-processing of signals using Gabor frames. Bemrose et al. [1] gave
a characterization of weaving frames (that does not require universal frame
bounds) and weaving Riesz bases. They proved a geometric characterization of
woven Riesz bases in terms of distance between subspaces of a Hilbert space.
Sufficient conditions for weaving frames by means of perturbation theory and
diagonal dominance can be found in [1]. The fundamental properties of weaving
frames reviewed by Casazza and Lynch in [2]. Casazza and Lynch [2] proved some
basic properties in the theory of weaving frames. They proved that an
invertible operator applied to woven frames leaves them woven. Casazza and
Lynch [2] considered a “weaving equivalent” of an unconditional basis for
weaving Riesz basis. Casazza, Freeman and Lynch [3] extended the concept of
weaving Hilbert space frames to the Banach space setting. They introduced and
studied _weaving Schauder frames_ in Banach spaces. It is proved in [3] that
for any two approximate Schauder frames for a Banach space $\mathcal{X}$,
every weaving is an approximate Schauder frame if and only if there is a
uniform constant $C\geq 1$ such that every weaving is a $C$-approximate
Schauder frame. Some perturbation theorems for woven approximate Schauder
frames can be found in [3]. Deepshikha and Vashisht studied weaving properties
of an infinite family of frames in separable Hilbert spaces in [11]. In [23,
22], authors introduced and studied weaving frames with respect to measure
spaces. Weaving properties of generalized frames and fusion frames can be
found in [9, 10, 21]. In [12], Dörfler and Faulhuber studied weaving Gabor
frames in $L^{2}(\mathbb{R})$. They gave a sufficient criteria for a family of
multi-window Gabor frames to be woven. A family of localization operators
related to weaving Gabor frames is also discussed in [12].

###  1.3. Outline of the paper

In this paper we give necessary and sufficient conditions for weaving
$K$-frames in Hilbert spaces. A characterization of weaving $K$-frames in
terms of a bounded linear operator is given, see Theorem 2.4. In Theorem 2.6,
a sufficient condition for $K$-frames not to be weakly woven in terms of lower
$K$-frame bounds is given. Theorem 2.8 shows that woven $K$-frames and weakly
woven $K$-frames are equivalent. A characterization of weaving $K$-frames in
terms of action of operators on $K$-frames is presented, see Proposition 2.10.
To be precise, it is shown that an operator $U\in\mathcal{B}(\mathcal{H})$
applied to woven $K$-frames leaves them $UK$-woven. A Paley-Wiener type
perturbation result for weaving $K$-frames is given in Theorem 2.13. Several
examples and counter-examples are given to illustrate the results.

##  2\. Main Results

We begin with the definition.

######  Definition 2.1.

A family of $K$-frames
$\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ for $\mathcal{H}$
is said to be _K-woven_ if there exist universal positive constants $A$ and
$B$ such that for any partition $\\{\sigma_{i}\\}_{i\in[m]}$ of $\mathbb{N}$,
the family $\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}}$ is a $K$-frame
for $\mathcal{H}$ with lower and upper $K$-frame bounds $A$ and $B$,
respectively. Each family $\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}}$
is called a weaving.

######  Definition 2.2.

A family of $K$-frames
$\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ for $\mathcal{H}$
is said to be _weakly K-woven_ if for any partition
$\\{\sigma_{i}\\}_{i\in[m]}$ of $\mathbb{N}$, the family
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}}$ is a $K$-frame for
$\mathcal{H}$.

In [1, Theorem 4.5], Bemrose et al. proved that if each weaving is a frame for
$\mathcal{H}$, then there exist uniform frame bounds $A_{o}\leq B_{o}<\infty$
that simultaneously work for all weavings. To be precise, weakly woven is
equivalent to the frames being woven. We extend this result for $K$-frame, see
Theorem 2.8. Our approach is different from the technique given in [1, Theorem
4.5], see Remark 2.9 for the details. In the direction of upper universal
bounds for ordinary frames, Bemrose et al. showed in [1, Proposition 3.1] that
every weaving of ordinary frames automatically has a universal upper frame
bound. Using a technique given in [1, Proposition 3.1], we have the following
result for universal upper frame bounds for $K$-frames.

######  Proposition 2.3.

For each $i\in[m]$, let $\\{\phi_{ij}\\}_{j=1}^{\infty}$ be a $K$-frame for
$\mathcal{H}$ with $K$-frame bounds $A_{i}$ and $B_{i}$. Then, for any
partition $\\{\sigma_{i}\\}_{i\in[m]}$ of $\mathbb{N}$, the family
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}}$ is a Bessel sequence with
Bessel bound $\sum_{i\in[m]}B_{i}$. That is, $\sum_{i\in[m]}B_{i}$ is one of
the choice for an universal upper $K$-frame bound.

###### Proof.

Let $\\{\sigma_{i}\\}_{i\in[m]}$ be any partition of $\mathbb{N}$. Then, for
any $f\in\mathcal{H}$, we have

$\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{i}}|\langle
f,\phi_{ij}\rangle|^{2}\leq\sum\limits_{i\in[m]}\sum\limits_{j\in\mathbb{N}}|\langle
f,\phi_{ij}\rangle|^{2}\leq\Big{(}\sum\limits_{i\in[m]}B_{i}\Big{)}\|f\|^{2}.$

This gives the required universal upper $K$-frame bound for the family
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}}$. ∎

The following theorem gives a necessary and sufficient condition for weaving
$K$-frames in terms of an operator.

######  Theorem 2.4.

For each $i\in[m]$, suppose $\\{\phi_{ij}\\}_{j=1}^{\infty}$ is a $K$-frame
for $\mathcal{H}$ with bounds $A_{i}$ and $B_{i}$. The following conditions
are equivalent.

  1. $(i)$

The family $\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ is
$K$-woven.

  2. $(ii)$

There exists $A>0$ such that for any partition
$\sigma=\\{\sigma_{i}\\}_{i\in[m]}$ of $\mathbb{N}$ there exists a bounded
linear operator $M_{\sigma}:\ell^{2}(\mathbb{N})\rightarrow\mathcal{H}$ such
that

$\displaystyle M_{\sigma}(e_{j})=$
$\displaystyle\begin{cases}\phi_{1j},j\in\sigma_{1},\\\
\phi_{2j},j\in\sigma_{2},\\\ \vdots\\\ \phi_{mj},j\in\sigma_{m},\end{cases}$

and $AKK^{*}\leq M_{\sigma}M_{\sigma}^{*}$, where $\\{e_{j}\\}_{j=1}^{\infty}$
is the canonical orthonormal basis for $\ell^{2}(\mathbb{N})$.

###### Proof.

$(i)\Rightarrow(ii):$ Suppose $A$ is an universal lower $K$-frame bound for
the family $\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$. For any
partition $\sigma=\\{\sigma_{i}\\}_{i\in[m]}$ of $\mathbb{N}$, let
$T_{\sigma}$ be the pre-frame operator associated with the Bessel sequence
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}}$.

Choose $M_{\sigma}=T_{\sigma}$. Then,
$M_{\sigma}(e_{j})=T_{\sigma}(e_{j})=\phi_{ij}$ for all $j\in\sigma_{i}$
$(i\in[m])$.
Next, we compute

$\displaystyle A\langle KK^{*}f,f\rangle$ $\displaystyle=A\|K^{*}f\|^{2}$
$\displaystyle\leq\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{i}}|\langle
f,\phi_{ij}\rangle|^{2}$ $\displaystyle=\sum\limits_{j\in\mathbb{N}}|\langle
f,M_{\sigma}(e_{j})\rangle|^{2}$
$\displaystyle=\sum\limits_{j\in\mathbb{N}}|\langle
M_{\sigma}^{*}f,e_{j}\rangle|^{2}$ $\displaystyle=\|M_{\sigma}^{*}f\|^{2}$
$\displaystyle=\langle M_{\sigma}M_{\sigma}^{*}f,f\rangle\ \text{for all}\
f\in\mathcal{H}.$

This gives $AKK^{*}\leq M_{\sigma}M_{\sigma}^{*}$.

$(ii)\Rightarrow(i)$ Let $\\{\sigma_{i}\\}_{i\in[m]}$ be any partition of
$\mathbb{N}$. Then, by using $(ii)$, for all $f\in\mathcal{H}$ we have

$\displaystyle A\|K^{*}f\|^{2}$ $\displaystyle=A\langle KK^{*}f,f\rangle$
$\displaystyle\leq\langle M_{\sigma}M_{\sigma}^{*}f,f\rangle$
$\displaystyle=\|M_{\sigma}^{*}f\|^{2}$
$\displaystyle=\sum\limits_{j\in\mathbb{N}}|\langle
M_{\sigma}^{*}f,e_{j}\rangle|^{2}$
$\displaystyle=\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{i}}|\langle
f,\phi_{ij}\rangle|^{2}.$

This gives the lower $K$-frame inequality. On the other hand, by Proposition
2.3 the positive number $\sum_{i\in[m]}B_{i}$ is one of the choice of an
universal upper $K$-frame bound. Hence, the family
$\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ is $K$-woven. ∎

Next we give an applicative example of Theorem 2.4.

######  Example 2.5.

Let $\mathcal{H}=\ell^{2}(\mathbb{N})$, $m=2$ and $\\{e_{j}\\}_{j=1}^{\infty}$
be the canonical orthonormal basis for $\mathcal{H}$.

  1. (a)

Define families $\Phi\equiv\\{\phi_{1j}\\}_{j=1}^{\infty}$ and
$\Psi\equiv\\{\phi_{2j}\\}_{j=1}^{\infty}$ in $\mathcal{H}$ as follows:

$\displaystyle\\{\phi_{1j}\\}_{j=1}^{\infty}$
$\displaystyle=\\{0,e_{2},0,e_{3},0,e_{4},0,\dots\\},$
$\displaystyle\\{\phi_{2j}\\}_{j=1}^{\infty}$
$\displaystyle=\\{0,e_{2},e_{2},e_{3},e_{3},e_{4},e_{4},\dots\\}.$

Let $K$ be the orthogonal projection of $\mathcal{H}$ onto
$[e_{j}]_{j=2}^{\infty}$. For any $\sigma\subset\mathbb{N}$, define a bounded
linear operator $M_{\sigma}:\ell^{2}(\mathbb{N})\rightarrow\mathcal{H}$ as
follows:

$\displaystyle M_{\sigma}(e_{j})=$
$\displaystyle\begin{cases}\phi_{1j},j\in\sigma,\\\
\phi_{2j},j\in\sigma^{c}.\end{cases}$

Then, for a given $f=\\{\alpha_{1},\alpha_{2},\dots\\}\in\mathcal{H}$ we have
$M^{*}_{\sigma}(f)=(0,\overline{\alpha_{2}},\widetilde{\alpha_{2}},\overline{\alpha_{3}},\widetilde{\alpha_{3}},\dots)$,
where $\overline{\alpha_{n}}$ the complex conjugate of the scalar $\alpha_{n}$
and $\widetilde{\alpha_{n}}$ is given by

$\displaystyle\widetilde{\alpha_{n}}=$ $\displaystyle\begin{cases}0,\
2n-1\in\sigma\quad(n\in\mathbb{N}),\\\ \overline{\alpha_{n}},\
2n-1\in\sigma^{c}\quad(n\in\mathbb{N}).\end{cases}$

Thus, for any $f=\\{\alpha_{1},\alpha_{2},\dots\\}\in\mathcal{H}$, we have

$\displaystyle\langle
M_{\sigma}M_{\sigma}^{*}f,f\rangle=\|M^{*}_{\sigma}f\|^{2}\geq\sum\limits_{i\geq
2}|\overline{\alpha_{i}}|^{2}=\|K^{*}f\|^{2}=\langle KK^{*}f,f\rangle.$

Hence, by Theorem 2.4, $\Phi$ and $\Psi$ are $K$-woven.

  2. (b)

Define families $\Phi_{1}\equiv\\{\phi_{1j}\\}_{j=1}^{\infty}$ and
$\Psi_{1}\equiv\\{\phi_{2j}\\}_{j=1}^{\infty}$ in $\mathcal{H}$ as follows:

$\displaystyle\\{\phi_{1j}\\}_{j=1}^{\infty}$
$\displaystyle=\\{e_{1},e_{2},0,e_{3},e_{4},e_{5},\dots\\},$
$\displaystyle\\{\phi_{2j}\\}_{j=1}^{\infty}$
$\displaystyle=\\{e_{1},0,e_{2},e_{3},e_{4},e_{5},\dots\\}.$

Let $K$ be the orthogonal projection of $\mathcal{H}$ onto
$[e_{j}]_{j=2}^{\infty}$. Then, $\Phi_{1}$ and $\Psi_{1}$ are $K$-frames for
$\mathcal{H}$.

Next we show that $\Phi_{1}$ and $\Psi_{1}$ are not $K$-woven. Choose
$\sigma=\mathbb{N}\setminus\\{2\\}$. Assume that there exists a bounded linear
operator $M_{\sigma}:\ell^{2}(\mathbb{N})\rightarrow\mathcal{H}$ such that

$\displaystyle M_{\sigma}(e_{j})=$ $\displaystyle\begin{cases}\phi_{1j},\
j\in\sigma,\\\ \phi_{2j},\ j\in\sigma^{c}.\end{cases}$

Then, for any $A>0$, we have

$\displaystyle A\langle KK^{*}e_{2},e_{2}\rangle$ $\displaystyle=A\langle
K^{*}e_{2},K^{*}e_{2}\rangle$ $\displaystyle=A\|e_{2}\|^{2}$ $\displaystyle>0$
$\displaystyle=\|M_{\sigma}e_{2}\|^{2}$ $\displaystyle=\langle
M_{\sigma}M_{\sigma}^{*}e_{2},e_{2}\rangle.$

Hence, by Theorem 2.4, $\Phi_{1}$ and $\Psi_{1}$ are not $K$-woven.

The following theorem provides a sufficient condition for $K$-frames not to be
weakly woven in terms of a lower $K$-frame bound. This is inspired by [1,
Lemma 4.3].

######  Theorem 2.6.

Suppose $\\{\phi_{ij}\\}_{j=1}^{\infty}$ is a K-frame for $\mathcal{H}$ with
bounds $A_{i}$ and $B_{i}$ $(i\in[m])$. Assume for any partition
$\\{\tau_{i}\\}_{i\in[m]}$ of a finite subset of $\mathbb{N}$ and for every
$A>0$ there exists a partition $\\{\sigma_{i}\\}_{i\in[m]}$ of
$\mathbb{N}\setminus\\{\tau_{i}\\}_{i\in[m]}$ such that
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}\cup\tau_{i}}$ has a lower
$K$-frame bound less than $A$. Then, there exists a partition
$\\{\pi_{i}\\}_{i\in[m]}$ of $\mathbb{N}$ such that
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\pi_{i}}$ is not a $K$-frame for
$\mathcal{H}$.

###### Proof.

Let $\tau_{1i}=\varnothing$ for all $i\in[m]$. Then, for $A_{1}=1$, there
exists a partition $\\{\sigma_{1i}\\}_{i\in[m]}$ of $\mathbb{N}$ such that
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{1i}\cup\tau_{1i}}$ has a lower
$K$-frame bound less than $1$. Therefore, there exists a vector
$h_{1}\in\mathcal{H}$ such that

$\displaystyle\sum\limits_{i\in[m]}$
$\displaystyle\sum\limits_{j\in\sigma_{1i}\cup\tau_{1i}}|\langle
h_{1},\phi_{ij}\rangle|^{2}<\|K^{*}h_{1}\|^{2}.$

Since $\sum\limits_{i\in[m]}\sum\limits_{j\in\mathbb{N}}|\langle
h_{1},\phi_{ij}\rangle|^{2}<\infty$, there exists $k_{1}\in\mathbb{N}$ such
that

$\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j>k_{1}}|\langle
h_{1},\phi_{ij}\rangle|^{2}<\|K^{*}h_{1}\|^{2}.$

Choose a partition $\\{\tau_{2i}\\}_{i\in[m]}$ of $[k_{1}]$ such that
$\tau_{2i}=\tau_{1i}\cup(\sigma_{1i}\cap[k_{1}])$ for all $i\in[m]$ and
$A_{2}=\frac{1}{2}$. Then, there exists a partition
$\\{\sigma_{2i}\\}_{i\in[m]}$ of
$\mathbb{N}\setminus\\{\tau_{2i}\\}_{i\in[m]}$ such that the family
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{2i}\cup\tau_{2i}}$ has a lower
$K$-frame bound less than $\frac{1}{2}$. Therefore, there exists a vector
$h_{2}\in\mathcal{H}$ such that

$\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{2i}\cup\tau_{2i}}|\langle
h_{2},\phi_{ij}\rangle|^{2}<\frac{1}{2}\|K^{*}h_{2}\|^{2}.$

By hypothesis, we have

$\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j\in\mathbb{N}}|\langle
h_{2},\phi_{ij}\rangle|^{2}<\infty.$

Therefore, there exists $k_{2}>k_{1}$ such that

$\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j>k_{2}}|\langle
h_{2},\phi_{ij}\rangle|^{2}<\frac{1}{2}\|K^{*}h_{2}\|^{2}.$

Continuing in this way, for $A_{n}=\frac{1}{n}$ and for a partition
$\\{\tau_{ni}\\}_{i\in[m]}$ of $[k_{n-1}]$ such that
$\tau_{ni}=\tau_{(n-1)i}\cup(\sigma_{(n-1)i}\cap[k_{n-1}])$ $(i\in[m])$, there
exists a partition $\\{\sigma_{ni}\\}_{i\in[m]}$ of
$\mathbb{N}\setminus\\{\tau_{ni}\\}_{i\in[m]}$ such that
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{ni}\cup\tau_{ni}}$ has a lower
$K$-frame bound less than $\frac{1}{n}$. Thus, there exists
$h_{n}\in\mathcal{H}$ such that

(2.1)
$\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{ni}\cup\tau_{ni}}|\langle
h_{n},\phi_{ij}\rangle|^{2}<\frac{1}{n}\|K^{*}h_{n}\|^{2},$

and we can find a positive integer $k_{n}>k_{n-1}$ such that

(2.2) $\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j>k_{n}}|\langle
h_{n},\phi_{ij}\rangle|^{2}<\frac{1}{n}\|K^{*}h_{n}\|^{2}.$

Choose a partition $\\{\pi_{i}\\}_{i\in[m]}$ of $\mathbb{N}$, where
$\pi_{i}=\bigcup_{n\in\mathbb{N}}\\{\tau_{ni}\\}$. We show that the family
$\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\pi_{i}}$ is not a $K$-frame for
$\mathcal{H}$. Assume that $\bigcup_{i\in[m]}\\{\phi_{ij}\\}_{j\in\pi_{i}}$ is
a $K$-frame for $\mathcal{H}$ with bounds $\alpha$ and $\beta$. By the
Archimedean property there exists a $\eta\in\mathbb{N}$ such that
$\eta>\frac{2}{\alpha}$.

Using (2.1) and (2.2), we compute

$\displaystyle\sum_{i\in[m]}\sum_{j\in\pi_{i}}|\langle
h_{\eta},\phi_{ij}\rangle|^{2}$
$\displaystyle\leq\sum_{i\in[m]}\sum_{j\in\tau_{(\eta+1)i}}|\langle
h_{\eta},\phi_{ij}\rangle|^{2}+\sum_{i\in[m]}\sum_{j\geq k_{\eta}}|\langle
h_{\eta},\phi_{ij}\rangle|^{2}$
$\displaystyle\leq\sum_{i\in[m]}\sum_{j\in\tau_{\eta i}\cup\sigma_{\eta
i}}|\langle h_{\eta},\phi_{ij}\rangle|^{2}+\sum_{i\in[m]}\sum_{j\geq
k_{\eta}}|\langle h_{\eta},\phi_{ij}\rangle|^{2}$
$\displaystyle\leq\frac{1}{\eta}\|K^{*}h_{\eta}\|^{2}+\frac{1}{\eta}\|K^{*}h_{\eta}\|^{2}$
$\displaystyle=\frac{2}{\eta}\|K^{*}h_{\eta}\|^{2}$
$\displaystyle<\alpha\|K^{*}h_{\eta}\|^{2},\ \text{a contradiction }.$

This completes the proof. ∎

Theorem 2.6 gives a necessary condition for weakly woven $K$-frames.

######  Proposition 2.7.

Suppose the family of $K$-frames
$\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ for $\mathcal{H}$
is weakly $K$-woven. Then, there exist a partition $\\{\tau_{i}\\}_{i\in[m]}$
of some finite subset of $\mathbb{N}$ and $A>0$ such that for any partition
$\\{\sigma_{i}\\}_{i\in[m]}$ of $\mathbb{N}\setminus\\{\tau_{i}\\}_{i\in[m]}$,
the family
$\bigcup\limits_{i\in[m]}\\{\phi_{ij}\\}_{j\in\sigma_{i}\cup\tau_{i}}$ has a
lower $K$-frame bound $A$.

Next we show that woven $K$-frames are equivalent to weakly woven $K$-frames.

######  Theorem 2.8.

Suppose $\Phi\equiv\\{\phi_{j}\\}_{j=1}^{\infty}$ and
$\Psi\equiv\\{\psi_{j}\\}_{j=1}^{\infty}$ are $K$-frames for $\mathcal{H}$.
The following are equivalent:

  1. $(i)$

$\Phi$ and $\Psi$ are $K$-woven.

  2. $(ii)$

$\Phi$ and $\Psi$ are weakly $K$-woven.

###### Proof.

$(i)\Rightarrow(ii):$ Obvious.

$(ii)\Rightarrow(i):$ First we note that an universal upper $K$-frame bound
for $\Phi$ and $\Psi$ can be obtained from Proposition 2.3. So it is
sufficient to compute an universal lower $K$-frame bound for $\Phi$ and
$\Psi$.

By Proposition 2.7, there exist disjoint finite sets $I$ and $J$ of
$\mathbb{N}$ and $A>0$ satisfying:

$(\maltese)$ For any partition $\\{\sigma,\delta\\}$ of
$\mathbb{N}\setminus(I\cup J)$, the family $\\{\phi_{j}\\}_{j\in
I\cup\sigma}\bigcup\\{\psi_{j}\\}_{j\in J\cup\delta}$ has a lower $K$-frame
bound $A$.

We can permute both the $K$-frames $\Phi$ and $\Psi$ (if necessary), so that
$I\cup J=[m]$.

Claim : For any partition $\\{I_{o},J_{o}\\}$ of $[m]$, there exists $A_{o}>0$
such that for any partition $\\{\sigma,\delta\\}$ of $\mathbb{N}\setminus[m]$
the family $\\{\phi_{j}\\}_{j\in I_{o}\cup\sigma}\bigcup\\{\psi_{j}\\}_{j\in
J_{o}\cup\delta}$ has a lower $K$-frame bound $A_{o}$. So that $\Phi$ and
$\Psi$ are $K$-woven with an universal lower $K$-frame bound
$\min\Big{\\{}{A_{o}:\\{I_{0},J_{0}\\}\text{ is a partition of}\
[m]}\Big{\\}}>0$.

Suppose our claim is not true. Then, there exists a partition
$\\{I_{1},J_{1}\\}$ of $[m]$ such that for each positive $\alpha$, there
exists a partition $\\{\sigma_{\alpha},\delta_{\alpha}\\}$ of
$\mathbb{N}\setminus[m]$ such that $\\{\phi_{j}\\}_{j\in
I_{1}\cup\sigma_{\alpha}}\bigcup\\{\psi_{j}\\}_{j\in
J_{1}\cup\delta_{\alpha}}$ has a lower $K$-frame bound less than $\alpha$.
Therefore, for any $n\in\mathbb{N}$, there exists a partition
$\\{\sigma_{n},\delta_{n}\\}$ of $\mathbb{N}\setminus[m]$ such that
$\\{\phi_{j}\\}_{j\in I_{1}\cup\sigma_{n}}\bigcup\\{\psi_{j}\\}_{j\in
J_{1}\cup\delta_{n}}$ has a lower $K$-frame bound less than $n$. Thus, there
exists $h_{n}\in\mathcal{H}$ such that

$\displaystyle\sum\limits_{j\in I_{1}\cup\sigma_{n}}|\langle
h_{n},\phi_{j}\rangle|^{2}+\sum\limits_{j\in J_{1}\cup\delta_{n}}|\langle
h_{n},\psi_{j}\rangle|^{2}$ $\displaystyle<\frac{1}{n}\|K^{*}(h_{n})\|^{2}$
$\displaystyle\leq\frac{1}{n}\lambda^{2}\|h_{n}\|^{2},\text{ where
}\lambda^{2}=\|K^{*}\|^{2}.$

That is

$\displaystyle\sum\limits_{j\in I_{1}\cup\sigma_{n}}|\langle\zeta
g_{n},\phi_{j}\rangle|^{2}+\sum\limits_{j\in J_{1}\cup\delta_{n}}|\langle\zeta
g_{n},\psi_{j}\rangle|^{2}<\frac{1}{n},$

where $\zeta=\frac{1}{\lambda}\ \text{and}\ g_{n}=\frac{h_{n}}{\|h_{n}\|}\
(n\in\mathbb{N})$. Note that $K^{*}(\zeta g_{n})\neq 0\ (n\in\mathbb{N})$.

Now we proceed in the following steps:

Step 1: By hypothesis, for each $n\in\mathbb{N}$ there exist a partition
$\\{\sigma_{n},\delta_{n}\\}$ of $\mathbb{N}\setminus[m]$ and a unit vector
$g_{n}\in\mathcal{H}$ such that

(2.3) $\displaystyle\sum\limits_{j\in I_{1}\cup\sigma_{n}}|\langle\zeta
g_{n},\phi_{j}\rangle|^{2}+\sum\limits_{j\in J_{1}\cup\delta_{n}}|\langle\zeta
g_{n},\psi_{j}\rangle|^{2}<\frac{1}{n}$

and the sets $\sigma_{n}$ and $\delta_{n}$ satisfy the following properties:

  1. (1)

For every $k=1,2,\dots$, either $m+k\in\sigma_{n}$ for all $n\geq k$ or
$m+k\in\delta_{n}$ for all $n\geq k$.

  2. (2)

There is a partition $\\{\sigma,\delta\\}$ of $\mathbb{N}\setminus[m]$ such
that $m+k\in\sigma$ implies that $m+k\in\sigma_{n}$ for all $n\geq k$ or if
$m+k\in\delta$ implies that $m+k\in\delta_{n}$ for all $n\geq k$.

Furthermore, the sequence $\\{\zeta g_{n}\\}_{n=1}^{\infty}$ is bounded, so
there exists a subsequence $\\{\zeta g_{n_{i}}\\}_{i=1}^{\infty}$ of $\\{\zeta
g_{n}\\}_{n=1}^{\infty}$ such that $\\{\zeta g_{n_{i}}\\}_{i=1}^{\infty}$
converges weakly to $h\in\mathcal{H}$. We reindex, $\zeta
g_{n_{i}}\rightarrow\zeta g_{i}$ and $\sigma_{n_{i}}\rightarrow\sigma_{i}$,
$\delta_{n_{i}}\rightarrow\delta_{i}$. Note that (2.3), (1) and (2) are
satisfied by this constructed sequence.

Step 2: In this step we show that $K^{*}(h)\neq 0$. Let
$\widetilde{K^{*}}:N(K^{*})^{\bot}\rightarrow R(K^{*})$ be the restriction of
$K^{*}$ on $N(K^{*})^{\bot}$. Note that $R(K^{*})$ is closed,
$\widetilde{K^{*}}$ is invertible and $\widetilde{K^{*}}^{-1}$ is bounded.
For any non-zero $f\in N(K^{*})^{\bot}$, $f=\widetilde{K^{*}}^{-1}(g)$ for
some $g\in R(K^{*})$.

We compute

(2.4) $\displaystyle\frac{\|K^{*}f\|}{\|f\|}$
$\displaystyle=\frac{\|\widetilde{K^{*}}f\|}{\|f\|}=\frac{\|\widetilde{K^{*}}(\widetilde{K^{*}}^{-1}g)\|}{\|\widetilde{K^{*}}^{-1}g\|}=\frac{\|g\|}{\|\widetilde{K^{*}}^{-1}g\|}\geq\frac{1}{\|\widetilde{K^{*}}^{-1}\|}.$

Fix $p\in\mathbb{N}$ such that
$p>\frac{2\lambda^{2}\|\widetilde{K^{*}}^{-1}\|^{2}}{A}$. So, we can find
$n_{p}\in\mathbb{N}$ such that for all $n\geq n_{p}>p$, we have

(2.5) $\displaystyle\sum\limits_{j\in[m+p]}|\langle\zeta
g_{n}-h,\phi_{j}\rangle|^{2}+\sum\limits_{j\in[m+p]}|\langle\zeta
g_{n}-h,\psi_{j}\rangle|^{2}<\frac{1}{2p}.$

From (✠), for each $n\in\mathbb{N}$ we have

(2.6) $\displaystyle\sum\limits_{j\in I\cup\sigma_{n}}|\langle\zeta
g_{n},\phi_{j}\rangle|^{2}+\sum\limits_{j\in J\cup\delta_{n}}|\langle\zeta
g_{n},\psi_{j}\rangle|^{2}\geq A\|K^{*}(\zeta g_{n})\|^{2}.$

For $m,p\in\mathbb{N}$, we write $[m,p]=\\{m+1,m+2,\dots,m+p\\}$. By using
(2.3), (2.4), (2.5), (2.6) and $\sigma\cap[m,p]=\sigma_{n_{p}}\cap[m,p]$,
$\delta\cap[m,p]=\delta_{n_{p}}\cap[m,p]$, we compute

$\displaystyle\sum\limits_{j\in I\cup\sigma}|\langle
h,\phi_{j}\rangle|^{2}+\sum\limits_{j\in J\cup\delta}|\langle
h,\psi_{j}\rangle|^{2}$ $\displaystyle\geq\sum\limits_{j\in
I\cup(\sigma\cap[m,p])}|\langle h,\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J\cup(\delta\cap[m,p])}|\langle h,\psi_{j}\rangle|^{2}$
$\displaystyle=\sum\limits_{j\in I\cup(\sigma_{n_{p}}\cap[m,p])}|\langle
h,\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J\cup(\delta_{n_{p}}\cap[m,p])}|\langle h,\psi_{j}\rangle|^{2}$
$\displaystyle\geq\frac{1}{2}\left(\sum\limits_{j\in
I\cup(\sigma_{n_{p}}\cap[m,p])}|\langle\zeta
g_{n_{p}},\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J\cup(\delta_{n_{p}}\cap[m,p])}|\langle\zeta
g_{n_{p}},\psi_{j}\rangle|^{2}\right)$ $\displaystyle-\left(\sum\limits_{j\in
I\cup(\sigma_{n_{p}}\cap[m,p])}|\langle h-\zeta
g_{n_{p}},\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J\cup(\delta_{n_{p}}\cap[m,p])}|\langle h-\zeta
g_{n_{p}},\psi_{j}\rangle|^{2}\right)$
$\displaystyle\geq\frac{1}{2}\left(\sum\limits_{j\in
I\cup\sigma_{n_{p}}}|\langle\zeta
g_{n_{p}},\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J\cup\delta_{n_{p}}}|\langle\zeta g_{n_{p}},\psi_{j}\rangle|^{2}\right)$
$\displaystyle-\frac{1}{2}\left(\sum\limits_{j\in\sigma_{n_{p}}\cap[m+p]^{c}}|\langle\zeta
g_{n_{p}},\phi_{j}\rangle|^{2}+\sum\limits_{j\in\delta_{n_{p}}\cap[m+p]^{c}}|\langle\zeta
g_{n_{p}},\psi_{j}\rangle|^{2}\right)$
$\displaystyle-\left(\sum\limits_{j\in[m+p]}|\langle h-\zeta
g_{n_{p}},\phi_{j}\rangle|^{2}+\sum\limits_{j\in[m+p]}|\langle h-\zeta
g_{n_{p}},\psi_{j}\rangle|^{2}\right)$
$\displaystyle\geq\frac{A}{2}\|K^{*}(\zeta
g_{n_{p}})\|^{2}-\frac{1}{2}\frac{1}{n_{p}}-\frac{1}{2p}$
$\displaystyle=\frac{A\zeta^{2}}{2}\|K^{*}(g_{n_{p}})\|^{2}-\frac{1}{2}\frac{1}{n_{p}}-\frac{1}{2p}$
$\displaystyle\geq\frac{A}{2\lambda^{2}\|\widetilde{K^{*}}^{-1}\|^{2}}-\frac{1}{p}$
$\displaystyle>0.$

This gives $h\neq 0$. Since $\\{\zeta g_{n}\\}_{n=1}^{\infty}\subset
N(K^{*})^{\bot}$ and $N(K^{*})^{\bot}$ is closed, so $K^{*}(h)\neq 0$.

Step 3: In this step we will show that $\\{\phi_{j}\\}_{j\in
I_{1}\cup\sigma}\bigcup\\{\psi_{j}\\}_{j\in J_{1}\cup\delta}$ is not a
$K$-frame for $\mathcal{H}$.
By using (2.3) and (2.5), we compute

$\displaystyle\sum\limits_{j\in I_{1}\cup\sigma}|\langle
h,\phi_{j}\rangle|^{2}+\sum\limits_{j\in J_{1}\cup\delta}|\langle
h,\psi_{j}\rangle|^{2}$
$\displaystyle=\lim_{p\rightarrow\infty}\left(\sum\limits_{j\in
I_{1}\cup(\sigma\cap[m,p])}|\langle h,\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J_{1}\cup(\delta\cap[m,p])}|\langle h,\psi_{j}\rangle|^{2}\right)$
$\displaystyle=\lim_{p\rightarrow\infty}\left(\sum\limits_{j\in
I_{1}\cup(\sigma_{n_{p}}\cap[m,p])}|\langle
h,\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J_{1}\cup(\delta_{n_{p}}\cap[m,p])}|\langle h,\psi_{j}\rangle|^{2}\right)$
$\displaystyle\leq 2\lim_{p\rightarrow\infty}\left(\sum\limits_{j\in
I_{1}\cup(\sigma_{n_{p}}\cap[m,p])}|\langle\zeta
g_{n_{p}},\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J_{1}\cup(\delta_{n_{p}}\cap[m,p])}|\langle\zeta
g_{n_{p}},\psi_{j}\rangle|^{2}\right)$
$\displaystyle+2\lim_{p\rightarrow\infty}\left(\sum\limits_{j\in
I_{1}\cup(\sigma_{n_{p}}\cap[m,p])}|\langle h-\zeta
g_{n_{p}},\phi_{j}\rangle|^{2}+\sum\limits_{j\in
J_{1}\cup(\delta_{n_{p}}\cap[m,p])}|\langle h-\zeta
g_{n_{p}},\psi_{j}\rangle|^{2}\right)$ $\displaystyle\leq
2\lim_{p\rightarrow\infty}\frac{1}{n_{p}}+2\lim_{p\rightarrow\infty}\frac{1}{2p}$
$\displaystyle=0$ $\displaystyle<\alpha\|K^{*}h\|^{2}\text{ for any
}\alpha>0.$

Therefore, $\\{\phi_{j}\\}_{j\in I_{1}\cup\sigma}\bigcup\\{\psi_{j}\\}_{j\in
J_{1}\cup\delta}$ is not a $K$-frame for $\mathcal{H}$. This contradicts the
fact that $\Phi$ and $\Psi$ are weakly $K$-woven. The proof is complete. ∎

######  Remark 2.9.

The technique used in the proof of Theorem 2.8 is different than given in [1,
Theorem 4.5] for an analogous result for ordinary frames in separable Hilbert
spaces. In the proof of Theorem 2.8, the operator $K$ is not invertible on the
whole space $\mathcal{H}$. So, computations in our proof are different than
given in [1, Theorem 4.5]. Furthermore, if $K=I$, the identity operator on
$\mathcal{H}$, then by Theorem 2.8, we can obtain [1, Theorem 4.5].

The following result characterizes weaving $K$-frames in terms of action of
operators on $K$-frames.

######  Proposition 2.10.

Let $\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ be a family of
$K$-frames for $\mathcal{H}$. The following are equivalent:

  1. $(i)$

$\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ is $K$-woven.

  2. $(ii)$

$\Big{\\{}\\{U(\phi_{ij})\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ is $UK$-woven
for all $U\in\mathcal{B}(\mathcal{H})$.

###### Proof.

$(i)\Rightarrow(ii):$ Let $A$ and $B$ be universal $K$-frame bounds for the
family $\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$. Let
$\\{\sigma_{i}\\}_{i\in[m]}$ be any partition of $\mathbb{N}$. Then, for any
$f\in\mathcal{H}$, we have

$\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{i}}|\langle
f,U(\phi_{ij})\rangle|^{2}=\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{i}}|\langle
U^{*}f,\phi_{ij}\rangle|^{2}\leq B\|U^{*}f\|^{2}\leq B\|U^{*}\|^{2}\|f\|^{2}.$

Similarly, for any $f\in\mathcal{H}$, we have

$\displaystyle\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{i}}|\langle
f,U(\phi_{ij})\rangle|^{2}=\sum\limits_{i\in[m]}\sum\limits_{j\in\sigma_{i}}|\langle
U^{*}f,\phi_{ij}\rangle|^{2}\geq A\|K^{*}U^{*}f\|^{2}=A\|(UK)^{*}f\|^{2}.$

Hence, the family
$\Big{\\{}\\{U(\phi_{ij})\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ is $UK$-woven
with universal $K$-frame bounds $A,\ B\|U^{*}\|^{2}$.

$(ii)\Rightarrow(i):$ Choose $U=I$, the identity operator on $\mathcal{H}$.
Then, the family $\Big{\\{}\\{\phi_{ij}\\}_{j=1}^{\infty}:i\in[m]\Big{\\}}$ is
$K$-woven. ∎

######  Remark 2.11.

Let $\Phi$ and $\Psi$ be $K$-frames for $\mathcal{H}$ such that $U\Phi$ and
$U\Psi$ are $UK$-woven for some $U\in\mathcal{B}(\mathcal{H})$. Then, in
general, $\Phi$ and $\Psi$ are not $K$-woven. This is justified in the
following example.

######  Example 2.12.

Let $\mathcal{H}=\ell^{2}(\mathbb{N})$ and let $K$ be the orthogonal
projection of $\mathcal{H}$ onto $[e_{j}]_{j=2}^{\infty}$. Define
$\Phi,\Psi\subset\mathcal{H}$ by

$\displaystyle\Phi\equiv\\{\phi_{1j}\\}_{j=1}^{\infty}$
$\displaystyle=\\{e_{1},0,e_{2},0,e_{3},0,e_{4},0,e_{5},0\dots\\},$
$\displaystyle\Psi\equiv\\{\phi_{2j}\\}_{j=1}^{\infty}$
$\displaystyle=\\{0,e_{1},0,e_{2},e_{3},e_{3},e_{4},e_{4},e_{5},e_{5},\dots\\},$

where $\\{e_{j}\\}_{j=1}^{\infty}$ is the canonical orthonormal basis for
$\mathcal{H}$. Then, $\Phi$ and $\Psi$ are $K$-frames for $\mathcal{H}$.

To show $\Phi$ and $\Psi$ are not $K$-woven. Choose
$\sigma=\mathbb{N}\setminus\\{1,3\\}$. Then, the family
$\\{\phi_{1j}\\}_{j\in\sigma}\bigcup\\{\phi_{2j}\\}_{j\in\sigma^{c}}=\\{0,0,0,0,e_{3},0,e_{4},0,e_{5},0,e_{6},\dots\\}$
is not a $K$-frame for $\mathcal{H}$, since for any $A>0$, we have

$\displaystyle\sum_{j\in\sigma}|\langle
e_{2},\phi_{1j}\rangle|^{2}+\sum_{j\in\sigma^{c}}|\langle
e_{2},\phi_{2j}\rangle|^{2}$ $\displaystyle=\sum_{j\geq 3}|\langle
e_{2},e_{j}\rangle|^{2}$ $\displaystyle=0$ $\displaystyle<A\|e_{2}\|^{2}$
$\displaystyle=A\|K^{*}e_{2}\|^{2}.$

Hence $\Phi$ and $\Psi$ are not $K$-woven.

Let $U$ be the orthogonal projection of $\mathcal{H}$ onto
$[e_{j}]_{j=3}^{\infty}$. Then, $U\Phi\equiv\\{U(\phi_{1j})\\}_{j=1}^{\infty}$
and $U\Psi\equiv\\{U(\phi_{2j})\\}_{j=1}^{\infty}$ are $UK$-woven frames for
$\mathcal{H}$. To show this, first we note that:

$\displaystyle\\{U(\phi_{1j})\\}_{j=1}^{\infty}$
$\displaystyle=\\{0,0,0,0,e_{3},0,e_{4},0,e_{5},0,e_{6},\dots\\},$
$\displaystyle\\{U(\phi_{2j})\\}_{j=1}^{\infty}$
$\displaystyle=\\{0,0,0,0,e_{3},e_{3},e_{4},e_{4},e_{5},e_{5},e_{6},\dots\\}.$

For any subset $\sigma$ of $\mathbb{N}$, we have

$\displaystyle\sum_{j\in\sigma}|\langle
f,U(\phi_{1j})\rangle|^{2}+\sum_{j\in\sigma^{c}}|\langle
f,U(\phi_{2j})\rangle|^{2}\leq 2\sum_{j=1}^{\infty}|\langle
f,e_{j}\rangle|^{2}=2\|f\|^{2},\ f\in\mathcal{H}.$

On the other hand, let $f\in\mathcal{H}$. Then,
$f=\sum\limits_{j=1}^{\infty}\alpha_{j}e_{j}$. Thus, we have

$\displaystyle\|(UK)^{*}f\|^{2}$ $\displaystyle=\|U^{*}f\|^{2}$
$\displaystyle=\|\sum\limits_{j\in\mathbb{N}}\alpha_{j}U^{*}e_{j}\|^{2}$
$\displaystyle=\|\sum\limits_{j=3}^{\infty}\alpha_{j}e_{j}\|^{2}$
$\displaystyle=\sum\limits_{j=3}^{\infty}|\langle f,e_{j}\rangle|^{2}$
$\displaystyle\leq\sum_{j\in\sigma}|\langle
f,U\phi_{1j}\rangle|^{2}+\sum_{j\in\sigma^{c}}|\langle
f,U\phi_{2j}\rangle|^{2}.$

Hence $U\Phi$ and $U\Psi$ are $UK$-woven with universal $UK$-frame bounds
$1,2$.

The next result gives a sufficient conditions for perturbation of weaving
$K$-frames.

######  Theorem 2.13.

Suppose $\Phi_{1}\equiv\\{\phi_{1j}\\}_{j=1}^{\infty}$ and
$\Phi_{2}\equiv\\{\phi_{2j}\\}_{j=1}^{\infty}$ are $K$-frames for
$\mathcal{H}$ with bounds $A_{1},\ B_{1}$ $($respectively, $A_{2},\ B_{2}$$)$
such that $\Phi_{1}$ is an orthogonal set and $\|\phi_{1j}\|^{2}>\alpha>0$ for
all $j\in\mathbb{N}$. Assume that there exists $\lambda,\mu,\nu\geq 0$ such
that

(2.7)
$\displaystyle\Big{(}\sqrt{B_{1}}+\sqrt{B_{2}}\Big{)}\left(\lambda+\mu\sqrt{B_{1}}+\nu\sqrt{B_{2}}\right)\|\widetilde{K^{*}}^{-1}\|<\sqrt{\alpha
A_{1}}$

and

(2.8)
$\displaystyle\Big{\|}\sum_{j=1}^{\infty}a_{j}(\phi_{1j}-\phi_{2j})\Big{\|}\leq\lambda\|\\{a_{j}\\}_{j=1}^{\infty}\|_{\ell^{2}(\mathbb{N})}+\mu\Big{\|}\sum_{j=1}^{\infty}a_{j}\phi_{1j}\Big{\|}+\nu\Big{\|}\sum_{j=1}^{\infty}a_{j}\phi_{2j}\Big{\|}$

for all sequence of scalars
$\\{a_{j}\\}_{j=1}^{\infty}\in\ell^{2}(\mathbb{N})$, where
$\widetilde{K^{*}}:N(K^{*})^{\bot}\rightarrow R(K^{*})$ is the restriction of
$K^{*}$ on $N(K^{*})^{\bot}$. Then, $\Phi_{1}$ and $\Phi_{2}$ are $K$-woven
with universal bounds $\frac{\Big{[}\sqrt{\alpha
A_{1}}-(\sqrt{B_{1}}+\sqrt{B_{2}})(\lambda+\mu\sqrt{B_{1}}+\nu\sqrt{B_{2}})\|\widetilde{K^{*}}^{-1}\|\Big{]}^{2}}{B_{1}+B_{2}}$
and $B_{1}+B_{2}$.

###### Proof.

Let $T^{i}$ be the pre-frame operator associated with $\Phi_{i},\ i=1,2$.
Then, $\|T^{i}\|\leq\sqrt{B_{i}}$. Let $\sigma$ be any subset of $\mathbb{N}$.
Define $T^{i}_{\sigma^{c}}:\ell^{2}(\mathbb{N})\rightarrow\mathcal{H}$ by

$\displaystyle
T^{i}_{\sigma^{c}}(\\{a_{j}\\}_{j=1}^{\infty})=\sum\limits_{j\in\sigma^{c}}a_{j}\phi_{ij},\
\\{a_{j}\\}_{j=1}^{\infty}\in\ell^{2}(\mathbb{N})\ \quad(i=1,2).$

Then, each $T^{i}_{\sigma^{c}}$ is a bounded operator with norm at most
$\sqrt{B_{i}}$. Indeed, for all
$\\{a_{j}\\}_{j=1}^{\infty}\in\ell^{2}(\mathbb{N})$, we have

$\displaystyle\left\|T^{i}_{\sigma^{c}}(\\{a_{j}\\}_{j\in\mathbb{N}})\right\|$
$\displaystyle=\Big{\|}\sum_{j\in\sigma^{c}}a_{j}\phi_{ij}\Big{\|}$
$\displaystyle\leq\|T^{i}\|\|\\{a_{j}\\}_{j=1}^{\infty}\|_{\ell^{2}(\mathbb{N})}$
$\displaystyle\leq\sqrt{B_{i}}\|\\{a_{j}\\}_{j=1}^{\infty}\|_{\ell^{2}(\mathbb{N})}\
\quad(i=1,2).$

By using (2.8), we compute

$\displaystyle\|(T^{1}_{\sigma^{c}}-T^{2}_{\sigma^{c}})\\{a_{j}\\}_{j\in\mathbb{N}}\|$
$\displaystyle=\Big{\|}\sum_{j\in\sigma^{c}}a_{j}(\phi_{1j}-\phi_{2j})\Big{\|}$
$\displaystyle=\Big{\|}\sum_{j\in\mathbb{N}}\gamma_{j}(\phi_{1j}-\phi_{2j})\Big{\|}\
(\text{where}\ \gamma_{j}=a_{j}\text{ for }j\in\sigma^{c}\ \text{and}\
\gamma_{j}=0\ \text{otherwise})$
$\displaystyle\leq\lambda\left\|\\{\gamma_{j}\\}_{j=1}^{\infty}\right\|_{\ell^{2}(\mathbb{N})}+\mu\Big{\|}\sum\limits_{j=1}^{\infty}\gamma_{j}\phi_{1j}\Big{\|}+\nu\Big{\|}\sum\limits_{j=1}^{\infty}\gamma_{j}\phi_{2j}\Big{\|}$
$\displaystyle\leq\lambda\left\|\\{\gamma_{j}\\}_{j=1}^{\infty}\right\|_{\ell^{2}(\mathbb{N})}+\mu\|T^{1}\|\left\|\\{\gamma_{j}\\}_{j=1}^{\infty}\right\|_{\ell^{2}(\mathbb{N})}+\nu\|T^{2}\|\left\|\\{\gamma_{j}\\}_{j=1}^{\infty}\right\|_{\ell^{2}(\mathbb{N})}$
(2.9)
$\displaystyle\leq\left(\lambda+\mu\sqrt{B_{1}}+\nu\sqrt{B_{2}}\right)\|\\{a_{j}\\}_{j=1}^{\infty}\|_{\ell^{2}(\mathbb{N})}\
\text{ for all }\\{a_{j}\\}_{j=1}^{\infty}\in\ell^{2}(\mathbb{N}).$

For any $f\in N(K^{*})^{\bot}$, by using (2) and orthogonality of $\Phi_{1}$,
we compute

$\displaystyle\Big{\|}\sum_{j\in\sigma}\langle
f,\phi_{1j}\rangle\phi_{1j}+\sum_{j\in\sigma^{c}}\langle
f,\phi_{2j}\rangle\phi_{2j}\Big{\|}$
$\displaystyle=\Big{\|}\sum_{j\in\sigma}\langle
f,\phi_{1j}\rangle\phi_{1j}+\sum_{j\in\sigma^{c}}\langle
f,\phi_{1j}\rangle\phi_{1j}-\sum_{j\in\sigma^{c}}\langle
f,\phi_{1j}\rangle\phi_{1j}+\sum_{j\in\sigma^{c}}\langle
f,\phi_{2j}\rangle\phi_{2j}\Big{\|}$
$\displaystyle=\Big{\|}\sum_{j\in\mathbb{N}}\langle
f,\phi_{1j}\rangle\phi_{1j}-\sum_{j\in\sigma^{c}}\langle
f,\phi_{1j}\rangle\phi_{1j}+\sum_{j\in\sigma^{c}}\langle
f,\phi_{2j}\rangle\phi_{2j}\Big{\|}$
$\displaystyle\geq\Big{\|}\sum_{j\in\mathbb{N}}\langle
f,\phi_{1j}\rangle\phi_{1j}\Big{\|}-\Big{\|}\sum_{j\in\sigma^{c}}\langle
f,\phi_{1j}\rangle\phi_{1j}-\sum_{j\in\sigma^{c}}\langle
f,\phi_{2j}\rangle\phi_{2j}\Big{\|}$
$\displaystyle\geq{\Big{(}\sum_{j\in\mathbb{N}}|\langle
f,\phi_{1j}\rangle|^{2}\|\phi_{1j}\|^{2}\Big{)}}^{\frac{1}{2}}-\Big{\|}\sum_{j\in\sigma^{c}}\langle
f,\phi_{1j}\rangle\phi_{1j}-\sum_{j\in\sigma^{c}}\langle
f,\phi_{1j}\rangle\phi_{2j}\Big{\|}$
$\displaystyle\quad-\Big{\|}\sum_{j\in\sigma^{c}}\langle
f,\phi_{1j}\rangle\phi_{2j}-\sum_{j\in\sigma^{c}}\langle
f,\phi_{2j}\rangle\phi_{2j}\Big{\|}$ $\displaystyle\geq\sqrt{\alpha
A_{1}}\|K^{*}(f)\|-\|T^{1}_{\sigma^{c}}T^{1*}(f)-T^{2}_{\sigma^{c}}T^{1*}(f)\|-\|T^{2}_{\sigma^{c}}T^{1*}(f)-T^{2}_{\sigma^{c}}T^{2*}(f)\|$
$\displaystyle=\sqrt{\alpha
A_{1}}\|K^{*}(f)\|-\|T^{1}_{\sigma^{c}}T^{1*}(\widetilde{K^{*}}^{-1}\widetilde{K^{*}}(f))-T^{2}_{\sigma^{c}}T^{1*}(\widetilde{K^{*}}^{-1}\widetilde{K^{*}}(f))\|$
$\displaystyle\quad-\|T^{2}_{\sigma^{c}}T^{1*}(\widetilde{K^{*}}^{-1}\widetilde{K^{*}}(f))-T^{2}_{\sigma^{c}}T^{2*}(\widetilde{K^{*}}^{-1}\widetilde{K^{*}}(f))\|$
$\displaystyle\geq\sqrt{\alpha
A_{1}}\|K^{*}(f)\|-\|T^{1*}\|\|T^{1}_{\sigma^{c}}-T^{2}_{\sigma^{c}}\|\|\widetilde{K^{*}}^{-1}\|\|\widetilde{K^{*}}(f)\|$
$\displaystyle\quad-\|T^{2}_{\sigma^{c}}\|\|T^{1*}-T^{2*}\|\|\widetilde{K^{*}}^{-1}\|\|\widetilde{K^{*}}(f)\|$
$\displaystyle\geq\sqrt{\alpha
A_{1}}\|K^{*}(f)\|-\sqrt{B_{1}}\left(\lambda+\mu\sqrt{B_{1}}+\nu\sqrt{B_{2}}\right)\|\widetilde{K^{*}}^{-1}\|\|K^{*}(f)\|$
$\displaystyle\quad-\sqrt{B_{2}}\left(\lambda+\mu\sqrt{B_{1}}+\nu\sqrt{B_{2}}\right)\|\widetilde{K^{*}}^{-1}\|\|K^{*}(f)\|$
(2.10) $\displaystyle=\Big{[}\sqrt{\alpha
A_{1}}-\Big{(}\sqrt{B_{1}}+\sqrt{B_{2}}\Big{)}\left(\lambda+\mu\sqrt{B_{1}}+\nu\sqrt{B_{2}}\right)\|\widetilde{K^{*}}^{-1}\|\Big{]}\|K^{*}(f)\|.$

Now, from Proposition 2.3, the family
$\\{\phi_{1j}\\}_{j\in\sigma}\bigcup\\{\phi_{2j}\\}_{j\in\sigma^{c}}$ is a
Bessel sequence in $\mathcal{H}$ with universal Bessel bound $B_{1}+B_{2}$.
Let $T$ be its associated pre-frame operator. Then, by using (2), we have

$\displaystyle\Big{[}\sqrt{\alpha
A_{1}}-\Big{(}\sqrt{B_{1}}+\sqrt{B_{2}}\Big{)}\left(\lambda+\mu\sqrt{B_{1}}+\nu\sqrt{B_{2}}\right)\|\widetilde{K^{*}}^{-1}\|\Big{]}^{2}\|K^{*}(f)\|^{2}$
$\displaystyle\leq\Big{\|}\sum_{j\in\sigma}\langle
f,\phi_{1j}\rangle\phi_{1j}+\sum_{j\in\sigma^{c}}\langle
f,\phi_{2j}\rangle\phi_{2j}\Big{\|}^{2}$ $\displaystyle=\|TT^{*}f\|^{2}$
$\displaystyle\leq\|T\|^{2}\|T^{*}f\|^{2}$
$\displaystyle\leq(B_{1}+B_{2})\Big{(}\sum_{j\in\sigma}|\langle
f,\phi_{1j}\rangle|^{2}+\sum_{j\in\sigma^{c}}|\langle
f,\phi_{2j}\rangle|^{2}\Big{)}\ \text{for all}\ f\in\mathcal{H}.$

Hence $\Phi_{1}$ and $\Phi_{2}$ are $K$-woven with the required universal
bounds. ∎

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