1 Introduction
The concept of Hilbert space frames was first introduced by Duffin and Schaeffer [Reference Duffin and Schaeffer12] in 1952. After several decades, in 1986, the importance of frame theory was popularized by the groundbreaking work by Daubechies, Grossman, and Meyer [Reference Daubechies, Grossmann and Meyer10]. Since then frame theory has been widely used by mathematicians and engineers in various fields of mathematics and engineering, namely, operator theory [Reference Han and Larson18], harmonic analysis [Reference Gröchenig16], wavelet analysis [Reference Gumber and Shukla17], signal processing [Reference Ferreira and Byrnes14], image processing [Reference Arefijamaal and Zekaee1], sensor network [Reference Casazza, Kutyniok, Li and Rozell8], data analysis [Reference Casazza and Kutyniok5], Retro Banach Frame [Reference Verma, Vashisht and Singh24], etc.
Frame theory literature became richer through several generalizations – fusion frame (frames of subspaces) [Reference Casazza and Kutyniok6, Reference Casazza, Kutyniok and Li7], G-frame (generalized frames) [Reference Sun23], K-frame (atomic systems) [Reference Gǎvruţa15], K-fusion frame (atomic subspaces) [Reference Bhandari and Mukherjee3], etc. - and these generalizations have been proved to be useful in various applications.
Over the years, weaving frames have been explored in various contexts (see [Reference Casazza, Freeman and Lynch4, Reference Deepshikha and Vashisht11, Reference Xu and Leng25]), yet their characterization through the structure of the associated kernel of the corresponding weaving synthesis operator remains largely unexplored. To bridge this gap, we undertake a comprehensive study of weaving frames and Riesz bases, focusing on their intrinsic properties and fundamental characterizations. Our approach provides a deeper understanding of their structural aspects, highlighting their stability, and robustness in different settings. By analyzing their interplay with synthesis operators, we establish new theoretical insights into their behavior. This work aims to contribute significantly to the ongoing research in frame theory and its applications in mathematical analysis.
Throughout this article,
$\mathcal {H}$
is a separable Hilbert space. We denote by
$\mathcal {L}(\mathcal {H}_1, \mathcal {H}_2)$
the space of all bounded linear operators from
$\mathcal {H}_1$
into
$\mathcal {H}_2$
, and
$\mathcal L({\mathcal H})$
for
$\mathcal L({\mathcal H}, {\mathcal H})$
. For
$T \in \mathcal {L}\mathcal {(H)}$
, we denote
$D(T), N(T),$
and
$R(T)$
for domain, null space and range of T, respectively, and
$\mathcal I$
is the identity operator.
2 Preliminaries
Before diving into the main sections, throughout this section, we recall basic definitions and results needed in this article. For a detailed discussion regarding frames, Riesz bases we refer to [Reference Christensen9].
2.1 Frame
A collection
$\{ f_i \}_{i\in I}$
in
$\mathcal {H}$
is called a frame if there exist constants
$A,B>0$
such that
$$ \begin{align} A\|f\|^2~ \leq ~\sum_{i\in I} |\langle f,f_i\rangle|^2 ~\leq ~B\|f\|^2, \end{align} $$
for all
$f \in \mathcal {H}$
. The numbers
$A, B$
are called frame bounds. The supremum over all A’s and infimum over all B’s satisfying above inequality are called the optimal frame bounds. If a collection satisfies only the right inequality in (2.1), it is called a Bessel sequence.
Given a frame
$\{f_i\}_{i\in I}$
for
$\mathcal {H}$
, the pre-frame operator or synthesis operator is a bounded linear operator
$T: l^2(I)\rightarrow \mathcal {H}$
and is defined by
$T\{c_i\}_{i \in I} = \sum \limits _{i\in I} c_i f_i$
. The adjoint of T,
$T^{*}: \mathcal {H} \rightarrow l^2(I)$
, given by
$T^{*}f = \{\langle f, f_i\rangle \}_{i \in I}$
, is called the analysis operator. The frame operator,
$S=TT^{*}: \mathcal {H}\rightarrow \mathcal {H}$
, is defined by
$$ \begin{align*}Sf=TT^{*}f = \sum_{i\in I} \langle f, f_i\rangle f_i.\end{align*} $$
It is well-known that the frame operator is bounded, positive, self adjoint, and invertible.
Here, we present the definition of weaving frames, which serves as the foundation for our study. This article primarily focuses on examining the structural properties and theoretical aspects of weaving frames.
Definition 2.1 [Reference Bemrose, Casazza, Gröchenig, Lammers and Lynch2]
Two frames
$\{f_n\}_{n =1}^{\infty }$
and
$\{g_n\}_{n =1}^{\infty }$
in a separable Hilbert space
${\mathcal H}$
are said to be weaving frames, if for every
$\sigma \subset \mathbb N$
,
$\{f_n\}_{n\in \sigma } \cup \{g_n\}_{n\in \sigma ^c}$
is a frame for
${\mathcal H}$
. i.e., for every
$f \in {\mathcal H}$
and for every
$\sigma \subset \mathbb N$
, there exist universal bounds
$\alpha \leq \beta $
so that we have the following inequality:
$$ \begin{align*}\alpha \|f\|^2\leq \sum\limits_{n \in \sigma} |\langle f, f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, g_n \rangle|^2 \leq \beta \|f\|^2.\end{align*} $$
3 Main results
In this section, we discuss various properties and characterizations of weaving frames and weaving Riesz bases.
Proposition 3.1 Let
$\mathcal F = \{f_n\}_{n=1}^{\infty }$
and
$\mathcal G = \{g_n\}_{n=1}^{\infty }$
be frames for
${\mathcal H}$
with bounds
$\alpha _1, \beta _1,$
and
$\alpha _2, \beta _2$
, and the associated synthesis operators
$T_{\mathcal F}$
and
$T_{\mathcal G}$
, respectively. Suppose
$0<\gamma <1$
so that
$\gamma (\|T_{\mathcal F}\| + \|T_{\mathcal G}\|) \leq \frac {\alpha _1}{2}$
. Further, if for every
$\{c_n\}_{n=1}^{\infty } \in \ell ^2$
, we have
$$ \begin{align} \|\sum\limits_{n=1}^{\infty} c_n(f_n - g_n)\| \leq \gamma \|\{c_n\}\|. \end{align} $$
Then
$\mathcal F$
and
$\mathcal G$
are weaving frames.
Proof For every
$\sigma \subset \mathbb N$
, let
$T_{\mathcal F_\sigma }(\{c_n\}_{n=1}^{\infty })=\sum \limits _{n \in \sigma } c_n f_n$
and
$T_{\mathcal G_\sigma }(\{c_n\}_{n=1}^{\infty })=\sum \limits _{n \in \sigma } c_n g_n$
.
Then applying [Theorem
$6.1$
, [Reference Bemrose, Casazza, Gröchenig, Lammers and Lynch2]] we have
$\|T_{\mathcal F_\sigma }\| \leq \|T_{\mathcal F}\|$
and
$\|T_{\mathcal G_\sigma }\| \leq \|T_{\mathcal G}\|$
. Using equation (3.1), we have
$$ \begin{align*}\|T_{\mathcal F_\sigma} - T_{\mathcal G_\sigma}\| \leq \|T_{\mathcal F} - T_{\mathcal G}\| <\gamma.\end{align*} $$
Therefore, for every
$f \in {\mathcal H}$
, we have
$$ \begin{align*} \|(T_{\mathcal F_\sigma} T^{*}_{\mathcal F_\sigma} - T_{\mathcal G_\sigma} T^{*}_{\mathcal G_\sigma})f\| &\leq \|T_{\mathcal F_\sigma}\| \| T^{*}_{\mathcal F_\sigma} - T^{*}_{\mathcal G_\sigma}\| \|f\| + \|T^{*}_{\mathcal G_\sigma}\| \| T_{\mathcal F_\sigma} - T_{\mathcal G_\sigma}\| \|f\| \\ & \leq (\|T_{\mathcal F}\| \| T_{\mathcal F} - T_{\mathcal G}\| + \|T_{\mathcal G}\| \| T_{\mathcal F} - T_{\mathcal G}\|)\|f\| \\ &\leq \gamma (\|T_{\mathcal F}\| + \|T_{\mathcal G}\|)\|f\| \\ &\leq \frac{\alpha_1}{2}\|f\|. \end{align*} $$
Thus, for every
$\sigma \subset \mathbb N$
and
$f \in {\mathcal H}$
, we obtain
$$ \begin{align*} &\left \|\sum\limits_{n \in \sigma} |\langle f, g_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, f_n \rangle|^2 \right\|\\ &= \left \|\sum\limits_{n=1}^{\infty} |\langle f, f_n \rangle|^2 + \left(\sum\limits_{n \in \sigma} |\langle f, g_n \rangle|^2 - \sum\limits_{n \in \sigma} |\langle f, f_n \rangle|^2\right)\right\|\\ &\geq \left \|\sum\limits_{n=1}^{\infty} |\langle f, f_n \rangle|^2 \right\| - \left \| \sum\limits_{n \in \sigma} |\langle f, g_n \rangle|^2 - \sum\limits_{n \in \sigma} |\langle f, f_n \rangle|^2 \right \|\\ &\geq \alpha_1 \|f\|^2 - \langle (T_{\mathcal G_\sigma} T^{*}_{\mathcal G_\sigma} - T_{\mathcal F_\sigma} T^{*}_{\mathcal F_\sigma})f, f \rangle\\ & \geq \alpha_1 \|f\|^2 - \frac{\alpha_1}{2}\|f\|^2\\ &= \frac{\alpha_1}{2}\|f\|^2. \end{align*} $$
On the other hand, the right-hand inequality will hold automatically. Consequently,
$\mathcal F$
and
$\mathcal G$
are weaving frames.
The following result provides necessary and sufficient conditions for the wovenness of the images of given weaving frames under a bounded linear operator.
Theorem 3.1
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
be two weaving frames for
${\mathcal H}$
with the universal bounds
$\alpha , \beta $
. If
$\mathcal T$
is a bounded linear operator in
${\mathcal H}$
. Then the following are equivalent:
-
(1) For every
$f \in {\mathcal H}$
, there exists
$\lambda>0$
so that
$\|\mathcal T^{*} f\|^2 \geq \lambda \|f\|^2$
. -
(2)
$\{\mathcal T f_n\}_{n=1}^{\infty }$
and
$\{\mathcal T g_n\}_{n=1}^{\infty }$
are weaving frames for
${\mathcal H}$
.
Proof (
$\underline {1 \implies 2}$
) Since
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are weaving frames for
${\mathcal H}$
with the universal bounds
$\alpha , \beta $
, then for every
$f \in {\mathcal H}$
and
$\sigma \subset \mathbb N$
, we have
$$ \begin{align} \alpha \|f\|^2\leq \sum\limits_{n \in \sigma} |\langle f, f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, g_n \rangle|^2 \leq \beta \|f\|^2. \end{align} $$
Again for every
$f \in {\mathcal H}$
, there exists
$\lambda =\inf \limits _{\|f\|=1} \|\mathcal T^{*}f\|^2>0$
for which we have
$\|\mathcal T^{*} f\|^2 \geq \lambda \|f\|^2$
. Therefore, applying equation (3.2), for every
$f \in {\mathcal H}$
and
$\sigma \subset \mathbb N$
, we have
$$ \begin{align*} \sum\limits_{n \in \sigma} |\langle f, \mathcal T f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, \mathcal T g_n \rangle|^2 &= \sum\limits_{n \in \sigma} |\langle \mathcal T^{*} f, f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle \mathcal T^{*} f, g_n \rangle|^2\\ & \geq \alpha \|\mathcal T^{*} f \|^2\\ & \geq \lambda ~\alpha \| f \|^2. \end{align*} $$
Furthermore, applying equation (3.2), for every
$f \in {\mathcal H}$
and
$\sigma \subset \mathbb N$
, we have
$$ \begin{align*} \sum\limits_{n \in \sigma} |\langle f, \mathcal T f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, \mathcal T g_n \rangle|^2 &= \sum\limits_{n \in \sigma} |\langle \mathcal T^{*} f, f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle \mathcal T^{*} f, g_n \rangle|^2\\ & \leq \beta \|\mathcal T^{*} f \|^2\\ & \leq \beta \|\mathcal T\|^2 \| f \|^2. \end{align*} $$
Thus,
$\{\mathcal T f_n\}_{n=1}^{\infty }$
and
$\{\mathcal T g_n\}_{n=1}^{\infty }$
are weaving frames for
${\mathcal H}$
.
(
$\underline {2 \implies 1}$
) This implication is very straightforward and follows directly from the given conditions.
The following result establishes a characterization of weaving frames using the associated hyperplane.
Theorem 3.2 Let
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
be frames for
${\mathcal H}$
. Then the following are equivalent:
-
(1)
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are weaving frames for
${\mathcal H}$
with the universal bounds
$\alpha , \beta $
. -
(2) For every
$\sigma \subset \mathbb N$
and every hyperplane
$\mathcal B \subset {\mathcal H}$
, we have
$$ \begin{align*}\alpha \leq \sum\limits_{n \in \sigma} \|P_{\mathcal B ^\perp}f_n\|^2 + \sum\limits_{n \in \sigma^c} \|P_{\mathcal B ^\perp}g_n\|^2 \leq \beta.\end{align*} $$
-
(3) For every
$\sigma \subset \mathbb N$
and every subspace
$\mathcal B_m$
with codimension m in
${\mathcal H}$
, we have
$$ \begin{align*}m \alpha \leq \sum\limits_{n \in \sigma} \|P_{\mathcal B_m ^\perp}f_n\|^2 + \sum\limits_{n \in \sigma^c} \|P_{\mathcal B_m ^\perp}g_n\|^2 \leq m \beta.\end{align*} $$
Proof (
$\underline {1 \implies 2}$
) Let
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
be weaving frames for
${\mathcal H}$
with the universal bounds
$\alpha , \beta $
. Then, for every
$f \in {\mathcal H}$
and
$\sigma \subset \mathbb N$
, we have
$$ \begin{align} \alpha \|f\|^2\leq \sum\limits_{n \in \sigma} |\langle f, f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, g_n \rangle|^2 \leq \beta \|f\|^2. \end{align} $$
For every
$f \in {\mathcal H}$
with
$\|f\|=1$
, let us assume that
$\mathcal B= (\text {span}\{f\})^\perp $
. Therefore, for every n, we obtain
$|\langle f, f_n \rangle |^2=\|P_{\mathcal B ^\perp }f_n\|^2$
and
$|\langle f, g_n \rangle |^2=\|P_{\mathcal B ^\perp }g_n\|^2$
. Thus, we have
$\alpha \geq \inf \left \{\sum \limits _{n \in \sigma } \|P_{\mathcal B ^\perp }f_n\|^2 + \sum \limits _{n \in \sigma ^c} \|P_{\mathcal B ^\perp }g_n\|^2: \mathcal B \text {~~is~~ a ~~hyperplane~~ in~~} {\mathcal H} \right \}$
and
$\beta \leq \sup \left \{\sum \limits _{n \in \sigma } \|P_{\mathcal B ^\perp }f_n\|^2 + \sum \limits _{n \in \sigma ^c} \|P_{\mathcal B ^\perp }g_n\|^2: \mathcal B \text {~~is~~ a ~~hyperplane~~ in~~} {\mathcal H} \right \}$
.
Applying equation (3.3), we obtain
$$ \begin{align*}\alpha \leq \sum\limits_{n \in \sigma} \|P_{\mathcal B ^\perp}f_n\|^2 + \sum\limits_{n \in \sigma^c} \|P_{\mathcal B ^\perp}g_n\|^2 \leq \beta.\end{align*} $$
(
$\underline {2 \implies 1}$
) Suppose
$\mathcal B$
is a hyperplane in
${\mathcal H}$
. Then, for every
$f \in \mathcal B^\perp $
with
$\|f\|=~1$
and every n, we have
$|\langle f, f_n \rangle |^2=\|P_{\mathcal B ^\perp }f_n\|^2$
and
$|\langle f, g_n \rangle |^2=\|P_{\mathcal B ^\perp }g_n\|^2$
. Thus, it is straightforward to verify that the assumed condition directly ensures that
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are weaving frames for
${\mathcal H}$
with the universal bounds
$\alpha , \beta $
.
(
$\underline {2 \implies 3}$
) Suppose for every
$\sigma \subset \mathbb N$
and every hyperplane
$\mathcal B \subset {\mathcal H}$
, we have
$$ \begin{align} \alpha \leq \sum\limits_{n \in \sigma} \|P_{\mathcal B ^\perp}f_n\|^2 + \sum\limits_{n \in \sigma^c} \|P_{\mathcal B ^\perp}g_n\|^2 \leq \beta. \end{align} $$
Let
$\mathcal B_m$
be a subspace of
${\mathcal H}$
with codimension m. Then
$\dim {\mathcal B_m}^{\perp } = m$
. Suppose
$\{e_1, e_2, \dots , e_m\}$
is an orthonormal basis in
${\mathcal B_m}^{\perp }$
and let us define
${\mathcal B_i}^{\perp } = \text {span} \{e_i\}$
, for every
$i=1, 2, \dots , m$
. Then using equation (3.4) with the hyperplane
$\mathcal B_i$
for every
$i = 1, 2, \ldots , m$
, we have
$$ \begin{align} \alpha \leq \sum\limits_{n \in \sigma} \|P_{\mathcal B_i ^\perp}f_n\|^2 + \sum\limits_{n \in \sigma^c} \|P_{\mathcal B_i ^\perp}g_n\|^2 \leq \beta. \end{align} $$
Furthermore, we have
$P_{\mathcal B_m ^\perp } = \sum \limits _{i=1}^m P_{\mathcal B_i ^\perp }$
. Thus, we obtain
$$ \begin{align*} &\sum\limits_{i=1}^m \left( \sum\limits_{n \in \sigma} \| P_{\mathcal B_i ^\perp} f_n\|^2 + \sum\limits_{n \in \sigma^c} \| P_{\mathcal B_i ^\perp} g_n\|^2 \right)\\ &= \sum\limits_{n \in \sigma} \sum\limits_{i=1}^m \| P_{\mathcal B_i ^\perp} f_n\|^2 + \sum\limits_{n \in \sigma^c} \sum\limits_{i=1}^m \| P_{\mathcal B_i ^\perp} g_n\|^2\\ &= \sum\limits_{n \in \sigma} \| P_{\mathcal B_m ^\perp} f_n\|^2 + \sum\limits_{n \in \sigma^c} \| P_{\mathcal B_m ^\perp} g_n\|^2. \end{align*} $$
Hence applying equation (3.5), we have
$$ \begin{align*}m \alpha \leq \sum\limits_{n \in \sigma} \|P_{\mathcal B_m ^\perp}f_n\|^2 + \sum\limits_{n \in \sigma^c} \|P_{\mathcal B_m ^\perp}g_n\|^2 \leq m \beta.\end{align*} $$
(
$\underline {3 \implies 2}$
) This implication is straightforward.
The following result provides a characterization of weaving frames through the associated weaving synthesis operator. In this theorem, we establish a necessary and sufficient condition for two frames to form weaving frames. This characterization is based on the structure of the image of the standard orthonormal basis in
$\ell ^2$
under the corresponding weaving synthesis operator. This characterization is achieved by examining how the frame vectors contribute to the structure of this image.
Theorem 3.3 Let
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
be two frames for
${\mathcal H}$
. Then the following are equivalent:
-
(1)
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are weaving frames for
${\mathcal H}$
. -
(2) For every
$\sigma \subset \mathbb N$
, if
$T_{\sigma }$
is the associated weaving synthesis operator, then we have where
$$ \begin{align*}T_{\sigma} e_n=\begin{cases} f_n : n \in \sigma\\ g_n : n \in \sigma^c \end{cases}\!\!\!\!\!\!\!,\end{align*} $$
$\{e_n\}_{n=1}^{\infty }$
is the canonical orthonormal basis in
$\ell ^2$
.
Proof (
$\underline {1 \implies 2}$
) Let
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
be weaving frames for
${\mathcal H}$
with the universal bounds
$0<\alpha \leq \beta < \infty $
. Then, for every
$\sigma \subset \mathbb N$
and for every
$f \in {\mathcal H}$
, we have
$$ \begin{align} \alpha \|f\|^2\leq \sum\limits_{n \in \sigma} |\langle f, f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, g_n \rangle|^2 \leq \beta \|f\|^2. \end{align} $$
Therefore, the operator
$T_{\sigma }^{*}: {\mathcal H} \rightarrow \ell ^2$
defined as
$$ \begin{align*}T_{\sigma}^{*} f=\{\langle f, f_n \rangle\}_{n \in \sigma} \cup \{\langle f, g_n \rangle\}_{n \in \sigma^c}\end{align*} $$
is bounded. Thus, for every
$f \in {\mathcal H}$
we have
$\|T_{\sigma }^{*} f\|^2 \geq \alpha \|f\|^2$
. Since for every
$f \in {\mathcal H}$
, we have
$$ \begin{align*}\langle f, T_{\sigma} e_n \rangle = \langle T_{\sigma}^{*} f, e_n \rangle=\begin{cases} \langle f, f_n \rangle : n \in \sigma\\ \langle f, g_n \rangle : n \in \sigma^c. \end{cases} \end{align*} $$
Therefore, we obtain
$T_{\sigma } e_n=\begin {cases} f_n : n \in \sigma \\ g_n : n \in \sigma ^c. \end {cases} $
(
$\underline {2 \implies 1}$
) Let
$T_{\sigma }$
be the associated weaving synthesis operator so that we have
$T_{\sigma } e_n=\begin {cases} f_n : n \in \sigma \\ g_n : n \in \sigma ^c. \end {cases} $
Therefore, for every
$f \in {\mathcal H}$
, we have
$$ \begin{align*}T_{\sigma}^{*}f = \sum\limits_{n=1}^{\infty} \langle T_{\sigma}^{*}f, e_n \rangle e_n = \sum\limits_{n\in \sigma} \langle f, f_n \rangle e_n + \sum\limits_{n\in \sigma^c} \langle f, g_n \rangle e_n.\end{align*} $$
Thus, for every
$f \in {\mathcal H}$
, we have
$$ \begin{align*} \sum\limits_{n \in \sigma} |\langle f, f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, g_n \rangle|^2 = \|T_{\sigma}^{*}f\|^2 \leq \|T_{\sigma}^{*}\|^2 \|f\|^2 \leq \beta \|f\|^2, \end{align*} $$
where
$\beta = \sup \limits _{\sigma } \|T_{\sigma }^{*}\|^2.$
Again since for every
$\sigma \subset \mathbb N$
, the associated synthesis operator
$T_{\sigma }$
is onto then
$T_{\sigma }^{*}$
is one-one with closed range (see [Reference Dunford and Schwartz13], p. 487). Therefore, it is bounded below on the unit sphere in
${\mathcal H}$
and hence there exists
$\alpha>0$
so that for every
$\sigma \subset \mathbb N$
and for every
$f \in {\mathcal H}$
, we obtain
$$ \begin{align*}\sum\limits_{n \in \sigma} |\langle f, f_n \rangle|^2 + \sum\limits_{n \in \sigma^c} |\langle f, g_n \rangle|^2 = \|T_{\sigma}^{*}f\|^2 \geq \alpha \|f\|^2.\end{align*} $$
Consequently,
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are weaving frames for
${\mathcal H}$
.
Let
${\mathcal H}_1, {\mathcal H}_2$
be two separable Hilbert spaces. It is well-known that, if
$\{f_n\}_{n=1}^{\infty }$
is a frame for
${\mathcal H}_1$
,
$\{g_n\}_{n=1}^{\infty }$
is a frame for
${\mathcal H}_2$
, and
$Q: {\mathcal H}_1 \rightarrow {\mathcal H}_2$
is an isomorphism so that
$Qf_n = g_n$
for every n, then the frames
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are equivalent (see [Reference Singer22]). Thus, equivalent frames preserve the same topological properties in the sense that their synthesis and analysis operators are related through the isomorphism Q, as their structural behavior remains invariant under equivalence relation. This concept is illustrated through the following example:
The frames
$\{(1, 0), (0, 1)\}$
and
$\{(2, 0), (0, 3)\}$
in
$\mathbb R^2$
are equivalent through
$Q=\begin {pmatrix} 2 & 0\\ 0 & 3\end {pmatrix} $
.
Let us consider an orthogonal projection
$P: \ell ^2 \rightarrow X$
, where X is a closed subspace of
$\ell ^2$
. Then
$\{Pe_n\}$
is a frame. If for every
$\sigma \subset \mathbb N$
,
$T_{\sigma }$
is the associated synthesis operator of two weaving frames
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
, and
$P: \ell ^2 \rightarrow (Ker T_{\sigma })^{\perp }$
is the orthogonal projection onto
$(Ker T_{\sigma })^{\perp }$
, then applying Theorem 3.3, we have
$$ \begin{align*} T_{\sigma} (P e_n)=\begin{cases} f_n : n \in \sigma\\ g_n : n \in \sigma^c. \end{cases} \end{align*} $$
Thus, for every
$\sigma \subset \mathbb N$
,
$T_{\sigma } : (Ker T_{\sigma })^{\perp } \rightarrow {\mathcal H}$
is an isometric onto map and hence we conclude the following remark.
Remark 3.1 Suppose
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are weaving frames and for every
$\sigma \subset \mathbb N$
, if
$T_{\sigma }$
is the associated weaving synthesis operator then the frames
$\{f_n\}_{n\in \sigma } \cup \{g_n\}_{n \in \sigma ^c}$
in
${\mathcal H}$
and
$\{P e_n\}_{n=1}^{\infty }$
in
$(Ker T_{\sigma })^{\perp }$
are equivalent with respect to the induced map
$T_{\sigma }$
.
Proof Since
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are weaving frames, then applying Theorem 3.3, for every
$\sigma \subset \mathbb N$
, we have
$$ \begin{align*} T_{\sigma} (P e_n)=\begin{cases} f_n : n \in \sigma\\ g_n : n \in \sigma^c, \end{cases} \end{align*} $$
where P is the orthogonal projection from
$\ell ^2$
onto
$(Ker T_{\sigma })^{\perp }$
. Thus,
$T_{\sigma }$
maps
$(Ker T_{\sigma })^{\perp }$
isometrically onto
${\mathcal H}$
.
Consequently, the frame
$\{f_n\}_{n\in \sigma } \cup \{g_n\}_{n \in \sigma ^c}$
in
${\mathcal H}$
is equivalent to the frame
$\{P e_n\}_{n=1}^{\infty }$
in
$(Ker T_{\sigma })^{\perp }$
(see [[Reference Singer22], p. 68]).
Definition 3.1 [Reference Holub19]
A sequence
$\{f_n\}_{n=1}^{\infty }$
is said to be a Besselian frame if it is a frame for
${\mathcal H}$
, and after the removal of a finite number of vectors, the resulting sequence forms a Riesz basis for
${\mathcal H}$
.
In this context it is to be noted that for a Besselian frame
$\{f_n\}_{n=1}^{\infty }$
in
${\mathcal H}$
,
$\sum \limits _{n=1}^{\infty } a_n f_n$
converges in
${\mathcal H}$
if and only if
$\{a_n\}_{n=1}^{\infty } \in \ell ^2.$
Example 3.1 If
$\{e_n\}_{n=1}^{\infty }$
is an orthonormal basis in
$\ell ^2$
, then
$\{e_1, e_1, e_2, e_3, e_4, \dots \}$
is a Besselian frame.
Definition 3.2 Two Besselian frames in
${\mathcal H}$
are said to be weaving Besselian frames, if every weaving of them is a Besselian frame for
${\mathcal H}$
.
Example 3.2 If
$\{e_n\}_{n=1}^{\infty }$
is an orthonormal basis in
$\ell ^2$
, then it is easy to verify that
$\{e_1, e_1, e_2, e_3, e_4, \dots \}$
and
$\{e_1, e_1, e_2, e_2, e_3, e_4, \dots \}$
are weaving Besselian frames.
Proposition 3.2 Let
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
be weaving Besselian frames for
${\mathcal H}$
with the associated weaving synthesis operator
$T_{\sigma } : \ell ^2 \rightarrow {\mathcal H}$
, for every
$\sigma \subset \mathbb N$
. Then the dimension of
$Ker(T_{\sigma })$
is finite.
Proof For every
$\sigma \subset \mathbb N$
, let us consider the orthogonal projection
$P: \ell ^2 \rightarrow (Ker T_{\sigma })^{\perp }$
. Then applying Remark 3.1,
$\{P e_n\}_{n=1}^{\infty }$
is a frame for
$(Ker T_{\sigma })^{\perp }$
and hence it is equivalent to the Besselian frame
$\{f_n\}_{n\in \sigma } \cup \{g_n\}_{n \in \sigma ^c}$
. Thus, applying [Theorem 2.3, [Reference Holub19, Theorem 2.3]], the dimension of
$Ker(T_{\sigma })$
is finite.
Motivated by the notion of Riesz basis, we discuss the concept of near Riesz basis in a similar framework. For detailed discussion on the same, we refer to the relevant literature cited herein [Reference Christensen9, Reference Singer22].
Definition 3.3 Let
$\{f_n\}_{n=1}^{\infty }$
be a frame for
${\mathcal H}$
. If there is a finite set
$\gamma $
so that
$\{f_n\}_{n \notin \gamma }$
is a Riesz basis in
${\mathcal H}$
, then
$\{f_n\}_{n=1}^{\infty }$
is said to be a near Riesz basis.
We define near weaving Riesz bases in analogy with the notion of weaving Besselian frames.
Definition 3.4 Let
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
be a two near Riesz bases in
${\mathcal H}$
. If there exists a finite subset
$\gamma $
in
$\mathbb N$
so that for every
$\sigma \subset {\mathbb N \setminus \gamma }$
,
$\{f_n\}_{n \in \sigma } \cup \{g_n\}_{n \in \sigma ^c}$
is a near Riesz basis in
${\mathcal H}$
, then
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are said to be near weaving Riesz bases.
Remark 3.2 It is to be noted that the concepts of Besselian frames and near Riesz bases are equivalent.
The following result provides a characterization of near weaving Riesz bases through the kernel of the associated weaving synthesis operator. This characterization offers insights into the interplay between the synthesis operator and the weaving property of near Riesz bases.
Theorem 3.4 Let
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
be weaving frames for
${\mathcal H}$
with the associated weaving synthesis operator
$T_{\sigma } : \ell ^2 \rightarrow {\mathcal H}$
, for every
$\sigma \subset \mathbb N$
. Then the following are equivalent:
-
(1)
$Ker T_{\sigma }$
is finite dimensional. -
(2)
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are near weaving Riesz bases in
${\mathcal H}$
.
Proof (
$\underline {1 \implies 2}$
) If for every
$\sigma \subset \mathbb N$
, P is the orthogonal projection from
$\ell ^2$
onto
$(Ker T_{\sigma })^{\perp }$
, then applying Remark 3.1 the frames
$\{P e_n\}_{n=1}^{\infty }$
and
$\{f_n\}_{n\in \sigma } \cup \{g_n\}_{n \in \sigma ^c}$
are equivalent. Therefore,
$\{f_n\}_{n\in \sigma } \cup \{g_n\}_{n \in \sigma ^c}$
is a near Riesz basis and hence
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are near weaving Riesz bases if and only if
$\{Pe_n\}_{n=1}^{\infty }$
is a near Riesz basis in
$(Ker T_{\sigma })^{\perp } = R(P)$
.
Let
$Ker T_{\sigma }$
is finite dimensional. Then
$(\mathcal I - P)$
represents the orthogonal projection onto
$Ker T_{\sigma }$
and hence we have
$\sum \limits _{n=1}^{\infty } \|(\mathcal I - P)e_n\|^2 < \infty $
(see [Reference Schatten21]). Therefore, there exists a positive integer k so that
$\sum \limits _{n=k+1}^{\infty } \|e_n - Pe_n\|^2 < 1$
.
Let us consider a sequence
$\{x_n\}_{n=1}^{\infty }$
where
$x_n=\begin {cases} e_n : n =1, 2, \dots , k\\ Pe_n : \text {otherwise}. \end {cases} $
Then we have
$\sum \limits _{n=1}^{\infty } \|e_n - x_n\|^2 < 1$
and hence
$\{x_n\}_{n=1}^{\infty }$
forms a basis in
$\ell ^2$
, which is equivalent to
$\{e_n\}_{n=1}^{\infty }$
(see [Reference Retherford and Holub20]). Therefore,
$\{x_n\}_{n=1}^{\infty }$
is a Riesz basis in
$\ell ^2$
and hence
$\{Pe_n\}_{n=k+1}^{\infty }$
is a Riesz basis in
$\overline {\text {span}} \{Pe_n\}_{n=k+1}^{\infty }$
inside
$\ell ^2$
.
Furthermore, since
$\overline {\text {span}} \{Pe_n\}_{n=1}^{\infty } = (Ker T_{\sigma })^{\perp }$
, then for every
$\sigma \subset \mathbb N$
,
$\{P e_n\}_{n=k+1}^{\infty }$
can be extended to a Riesz basis in
$(Ker T_{\sigma })^{\perp }$
. Hence, there exists
$\gamma \subset \{1, 2, \dots , k\}$
for which
$\{P e_n\}_{n\notin \gamma }$
is a Riesz basis in
$(Ker T_{\sigma })^{\perp }$
. Thus,
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are near weaving Riesz bases in
${\mathcal H}$
.
(
$\underline {2 \implies 1}$
) If
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are near weaving Riesz bases, then the removal of finite number of elements from every weaving ensures that the remaining sequence forms a Riesz basis. Thus, it is easy to verify that
$\{f_n\}_{n=1}^{\infty }$
and
$\{g_n\}_{n=1}^{\infty }$
are Besselian weaving frames.
Therefore, applying Proposition 3.2 for every
$\sigma \subset \mathbb N$
,
$Ker T_{\sigma }$
is finite dimensional.
