On Weaving Generalized Frames and Generalized Riesz Bases

Abstract

Weaving frames have potential applications in wireless sensor networks that require distributed processing of signals under different frames. In this paper, we study some new properties of weaving generalized frames (or g-frames) and weaving generalized orthonormal bases (or g-orthonormal bases). It is shown that a g-frame and its dual g-frame are woven. The inter-relation of optimal g-frame bounds and optimal universal g-frame bounds is studied. Further, we present a characterization of weaving g-frames. Illustrations are given to show the difference in properties of weaving generalized Riesz bases and weaving Riesz bases.

1 Introduction

The concept of frames was introduced by Duffin and Schaeffer [11] and popularized by Daubechies et al. [7] when they showed the importance of frames in data processing. Frames can view as the generalization of bases but allow for over completeness. This redundancy of frames makes them play a vital role in numerous areas viz., noise reduction, sparse representations, image compression, signal transmission and processing, image processing, wavelet analysis, etc. Frames also help to spread the information over a wider range of vectors and thus it provides resilience against losses or noises. For basic theory and applications of frames, we refer [3, 6, 8, 12, 14].

The concept of generalized frames or g-frames was introduced by Sun [18]. G-frames are generalization of many frames like ordinary frames, frames of subspaces, pseudo-frames, etc. These frames are useful in many applications. In this paper, we study weaving generalized frames and weaving generalized Riesz bases. The notion of weaving frames was introduced by Bemrose et al. [1]. Weaving frames have potential applications in wireless sensor networks that require distributed signal processing under different frames, as well as preprocessing of signals using Gabor frames.

2 Preliminaries

Throughout the paper, $\mathcal {H}$ is a separable Hilbert space and $\{\mathcal {H}_m\}_{m \in \mathbb {N}}$ is a sequence of subspaces of a separable Hilbert space. $\mathcal {L}(\mathcal {H}, \mathcal {H}_m)$ is the space of all linear bounded operators from $\mathcal {H}$ to $\mathcal {H}_m$. If $U \in \mathcal {L}(\mathcal {H}, \mathcal {K})$ then $U^*$ denotes the Hilbert-adjoint operator of U, where $\mathcal {H}$ and $\mathcal {K}$ are Hilbert spaces. w.r.t. is the abbreviation used for with respect to.

2.1 Frames

Suppose $\{h_m\}_{m\in \mathbb {N}}$ is a countable sequence of vectors in $\mathcal {H}$. Then, $\{h_m\}_{m\in \mathbb {N}}$ is called a frame (or ordinary frame) for $\mathcal {H}$ if there exist positive constants $A\le B$ such that for any $h\in \mathcal {H}$,

$$\begin{aligned} A \Vert h\Vert ^2\le \sum _{m \in \mathbb {N}} |\langle h, h_m\rangle |^2 \le B \Vert h\Vert ^2. \end{aligned}$$

(1)

If $\{h_m\}_{m\in \mathbb {N}}$ satisfies only upper inequality in (1), then it is called a Bessel sequence and B is called a Bessel bound.

If a frame $\{h_m\}_{m\in \mathbb {N}}$ ceases to be a frame when an arbitrary element is removed then $\{h_m\}_{m\in \mathbb {N}}$ is called an exact frame.

Associated with a Bessel sequence $\{h_m\}_{m\in \mathbb {N}}$, the frame operator $S:\mathcal {H}\rightarrow \mathcal {H}$ is defined by

$$\begin{aligned} S(h)=\sum _{m \in \mathbb {N}}\langle h, h_m \rangle h_m . \end{aligned}$$

The frame operator S is linear, bounded and self-adjoint. If the Bessel sequence $\{h_m\}_{m\in \mathbb {N}}$ is a frame then the frame operator is invertible. Using the frame operator, we have a series representation of each vector $h\in \mathcal {H}$ in terms of frame elements which is given by

$$\begin{aligned}&h = \sum _{m \in \mathbb {N}} \langle h, S^{-1}h_m \rangle h_m \ =\sum _{m \in \mathbb {N}}\langle h,h_m\rangle S^{-1}h_m . \end{aligned}$$

If a frame $\{h_m\}_{m\in \mathbb {N}}$ is not exact then there exist a frame $\{g_m\}_{m\in \mathbb {N}}$ other than $\{S^{-1}h_m\}_{m\in \mathbb {N}}$ such that

$$\begin{aligned}&h = \sum _{m \in \mathbb {N}} \langle h, g_m \rangle h_m,\ \text {for all } h\in \mathcal {H}. \end{aligned}$$

Here, $\{g_m\}_{m\in \mathbb {N}}$ is called a dual frame of $\{h_m\}_{m\in \mathbb {N}}$. Thus, a frame can provide more than one series representation of a vector in terms of the frame elements.

A sequence $\{h_m\}_{m\in \mathbb {N}}\subset \mathcal {H}$ is called a Riesz basis for $\mathcal {H}$ if $\{h_m\}_{m\in \mathbb {N}}$ is complete in $\mathcal {H}$, and there exist positive constants $A\le B$ such that for any finite scalar sequence $\{c_m\}$,

$$\begin{aligned} A\sum |c_m|^2\le \left\| \sum c_m h_m\right\| ^2\le B\sum |c_m|^2. \end{aligned}$$

Riesz bases are the images of orthonormal bases under bounded invertible operators [6]. Thus, these can be viewed as generalization of orthonormal bases.

2.2 Weaving Frames

We start with the definition of weaving frames which was given by Bemrose et al. [1].

Definition 2.1

Frames $\{\phi _{m}\}_{m\in \mathbb {N}}$ and $\{\psi _{m}\}_{m\in \mathbb {N}}$ for $\mathcal {H}$ are called woven if there exist positive constants $A\le B$ such that for any $\sigma \subseteq \mathbb {N}$, $\{\phi _{m}\}_{m\in \sigma }\cup \{\psi _{m}\}_{m\in \sigma ^c}$ is a frame for $\mathcal {H}$ with lower frame bound A and upper frame bound B. Each $\{\phi _{m}\}_{m\in \sigma }\cup \{\psi _{m}\}_{m\in \sigma ^c}$ is called a weaving.

Bemrose et al. presented one interesting result in [1] which says that a Riesz basis and a frame (which is not a Riesz basis) are not woven.

Theorem 2.1

[1] Suppose $\{\phi _{m}\}_{m\in \mathbb {N}}$ is a Riesz basis and $\{\psi _{m}\}_{m\in \mathbb {N}}$ is a frame for $\mathcal {H}$. If $\{\phi _{m}\}_{m\in \mathbb {N}}$ and $\{\psi _{m}\}_{m\in \mathbb {N}}$ are woven, then $\{\psi _{m}\}_{m\in \mathbb {N}}$ is a Riesz basis.

Following result presented in [1] says that if two Riesz bases are woven, then every weaving is a Riesz basis.

Theorem 2.2

[1] Suppose $\{\phi _{m}\}_{m\in \mathbb {N}}$ and $\{\psi _{m}\}_{m\in \mathbb {N}}$ are Riesz bases for $\mathcal {H}$ and there is a uniform constant $A>0$ so that for any $\sigma \subset \mathbb {N}$, $\{\phi _{m}\}_{m\in \sigma }\cup \{\psi _{m}\}_{m\in \sigma ^c}$ is a frame with lower frame bound A. Then, for any $\sigma \subset \mathbb {N}$, $\{\phi _{m}\}_{m\in \sigma }\cup \{\psi _{m}\}_{m\in \sigma ^c}$ is a Riesz basis.

Many interesting properties of weaving frames were studied by Casazza et al. [5]. Then, the notion of weaving frames in Hilbert spaces was extended to Banach spaces in [2]. A characterization for the weaving of approximate Schauder frames in terms of C-approximate Schauder frame was presented. The concept of weaving frames in different settings was studied by many authors in [9, 10, 15, 19, 20].

2.3 Generalized Frames in Hilbert Spaces

Sun [18] gave the concept of generalized frames or g-frames which is the generalization of ordinary frames, fusion frames, bounded quasi-projectors, etc., see [4, 13, 16, 17].

Definition 2.2

[18] A sequence $\{\varLambda _m\in \mathcal {L}(\mathcal {H},\mathcal {H}_m):m \in \mathbb {N}\}$ is a generalized frame (or g-frame) for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$ if there exist positive constants $A\le B$ such that

$$\begin{aligned} A\Vert h\Vert ^2 \le \sum \limits _{m\in \mathbb {N}}\Vert \varLambda _m h\Vert ^2 \le B\Vert h\Vert ^2,\ \text {for all } h\in \mathcal {H}. \end{aligned}$$

(2)

The constants A and B are called lower and upper g-frame bounds, respectively. The supremum of all lower g-frame bounds is called the optimal lower g-frame bound, and the infimum of all upper g-frame bounds is called the optimal upper g-frame bound.

If $\{\varLambda _m\}_{m\in \mathbb {N}}$ satisfies the upper inequality in (2) then it is called a g-Bessel sequence for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$ and B is called a g-Bessel bound.

If a g-frame $\{\varLambda _m\}_{m\in \mathbb {N}}$ ceases to be a g-frame when an arbitrary element is removed then $\{\varLambda _m\}_{m\in \mathbb {N}}$ is called a g-exact frame.

Associated with a g-frame $\{\varLambda _m\}_{m\in \mathbb {N}}$, the g-frame operator $S:\mathcal {H}\rightarrow \mathcal {H}$ is defined by

$$\begin{aligned} S(h) =\sum _{m\in \mathbb {N}}\varLambda ^*_m \varLambda _m h. \end{aligned}$$

The g-frame operator S is linear, bounded, self-adjoint and invertible.

Definition 2.3

[18] Suppose $\{\varLambda _m\}_{m\in \mathbb {N}}$ and $\{\varGamma _m\}_{m\in \mathbb {N}}$ are g-frames for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$ such that

$$\begin{aligned} h=\sum _{m\in \mathbb {N}}\varLambda _m^*\varGamma _m h=\sum _{m\in \mathbb {N}}\varGamma _m^*\varLambda _m h,\ \text {for all } h\in \mathcal {H}. \end{aligned}$$

Then, $\{\varGamma _m\}_{m\in \mathbb {N}}$ is called a dual g-frame of $\{\varLambda _m\}_{m\in \mathbb {N}}$.

Sun [18] introduced the concept of generalized Riesz basis or g-Riesz basis which is the generalization of Riesz basis.

Definition 2.4

[18] A sequence $\{\varLambda _m\in \mathcal {L}(\mathcal {H}, \mathcal {H}_m):m \in \mathbb {N}\}$ is called a generalized Riesz basis (or g-Riesz basis) for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$ if

• (i) $\{\varLambda _m\}_{m\in \mathbb {N}}$ is complete in $\mathcal {H}$ that is $\{h:\varLambda _m h=0, m\in \mathbb {N}\}=\{0\}$

• (ii) There exist positive constants $A\le B$ such that for any finite set $\mathcal {J}\subset \mathbb {N}$ and $h_m\in \mathcal {H}_m$,

  $$\begin{aligned} A\sum _{m\in \mathcal {J}}\Vert h_m\Vert ^2\le \Big \Vert \sum _{m\in \mathcal {J}}\varLambda _m^*h_m\Big \Vert ^2\le B\sum _{m\in \mathcal {J}}\Vert h_m\Vert ^2. \end{aligned}$$

The constants A and B are called lower and upper g-Riesz bounds, respectively.

Definition 2.5

[18] A sequence $\{\varLambda _m\in \mathcal {L}(\mathcal {H}, \mathcal {H}_m):m \in \mathbb {N}\}$ is called a g-orthonormal basis for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$ if

• (i) $\langle \varLambda _{m_1}^*h_{m_1}, \varLambda _{m_2}^*h_{m_2}\rangle =\delta _{m_1,m_2}\langle h_{m_1}, h_{m_2}\rangle$, $\ \text {for all } m_1,m_2\in \mathbb {N},\ h_{m_1}\in \mathcal {H}_{m_1},\ h_{m_2}\in \mathcal {H}_{m_2}$

• (ii) $\sum \nolimits _{m\in \mathbb {N}}\Vert \varLambda _mh\Vert ^2=\Vert h\Vert ^2$ $,\ \text {for all } h\in \mathcal {H}$.

Sun [18] characterized g-frames, g-orthonormal bases and g-Riesz bases using orthonormal basis for $\mathcal {H}_m$.

Theorem 2.3

[18] Let $\varLambda _m\in \mathcal {L}(\mathcal {H}, \mathcal {H}_m)$and $\{e_{n,m}\}_{n\in J_m}$ be an orthonormal basis for $\mathcal {H}_m$, where $J_m\subseteq \mathbb {N}$, $m \in \mathbb {N}$. Then, $\{\varLambda _m\}_{m\in \mathbb {N}}$ is a g-frame (respectively, g-Riesz basis, g-orthonormal basis) for $\mathcal {H}$ if and only if $\{\varLambda _m^*e_{n,m}\}_{n\in J_m,m\in \mathbb {N}}$ is a frame (respectively, Riesz basis, orthonormal basis) for $\mathcal {H}$.

3 Weaving Generalized Frames

We begin this section with the definition of weaving g-frames in separable Hilbert spaces.

Definition 3.1

[15] Two g-frames $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_{m} \}_{m\in \mathbb {N}}$ are called woven if there exist positive constants $A\le B$ such that for any $\sigma \subseteq \mathbb {N}$, $\{\varLambda _{m}\}_{m\in \sigma }\cup \{\varOmega _{m}\}_{m\in \sigma ^c}$ is a g-frame for $\mathcal {H}$ with lower g-frame bound A and upper g-frame bound B.

The constants A and B are called universal lower g-frame bound and universal upper g-frame bound, respectively. The supremum of all universal lower g-frame bounds is called the optimal universal lower g-frame bound, and the infimum of all upper g-frame bounds is called the optimal universal upper g-frame bound.

The following proposition gives the existence of universal upper g-frame bound for any two g-frames $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$.

Proposition 3.1

[20] Suppose $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are g-frames for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_{m}\}_{m\in \mathbb {N}}$ with upper g-frame bounds $B_1$ and $B_2$, respectively. Then, $B_1+B_2$ is a universal upper g-frame bound of $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$.

In Theorem 2.3, g-frames are characterized using orthonormal basis for $\mathcal {H}_m$. Following theorem is an extension of Theorem 2.3 but it characterizes weaving g-frames using frames for $\mathcal {H}_m$ instead of orthonormal bases.

Theorem 3.1

Let $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ be g-frames for $\mathcal {H}$ w.r.t. $\{ \mathcal {H}_m\}_{m\in \mathbb {N}}$. Suppose $\{f_{n,m}\}_{n\in J_m}$ and $\{g_{n,m}\}_{n\in J_m}$ are frames for $\mathcal {H}_m$ with lower frame bounds $A_{1,m}$, $A_{2,m}$ (respectively) and upper frame bounds $B_{1,m}$, $B_{2,m}$ (respectively). If there exist positive constants $A_1<B_1$ and $A_2<B_2$ such that $0<A_1\le A_{1,m}\le B_{1,m}\le B_1<\infty$ and $0<A_2\le A_{2,m}\le B_{2,m}\le B_2<\infty$, for all $m\in \mathbb {N}$, then $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ if and only if $\{\varLambda _{m}^*f_{n,m}\}_{n\in J_m, m\in \mathbb {N}}$ and $\{\varOmega _{m}^*g_{n,m}\}_{n\in J_m, m\in \mathbb {N}}$ are weaving frames for $\mathcal {H}$.

Proof

First suppose that $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ with universal lower and upper g-frame bounds A and B, respectively.

Let $\sigma$ be any subset of $\mathbb {N}$. Then, for any $h\in \mathcal {H}$, we compute

$$\begin{aligned} A\Vert h\Vert ^2&\le \sum _{m\in \sigma } \Vert \varLambda _{m} h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _m h\Vert ^2\\&\le \sum _{m\in \sigma } \frac{1}{A_{1,m}}\sum _{n\in J_m}|\langle \varLambda _{m} h,f_{n,m}\rangle |^2+\sum _{m\in \sigma ^c}\frac{1}{A_{2,m}}\sum _{n\in J_m}|\langle \varOmega _m h,g_{n,m}\rangle |^2\\&\le \frac{1}{A_1}\sum _{m\in \sigma }\sum _{n\in J_m}|\langle h,\varLambda _{m}^*f_{n,m}\rangle |^2+\frac{1}{A_{2}}\sum _{m\in \sigma ^c}\sum _{n\in J_m}| h,\varOmega _m^*g_{n,m}\rangle |^2\\&\le \max \left\{ \frac{1}{A_1},\frac{1}{A_{2}}\right\} \left( \sum _{m\in \sigma }\sum _{n\in J_m}|\langle h,\varLambda _{m}^*f_{n,m}\rangle |^2+\sum _{m\in \sigma ^c}\sum _{n\in J_m}| h,\varOmega _m^*g_{n,m}\rangle |^2\right) . \end{aligned}$$

Similarly,

$$\begin{aligned} B\Vert h\Vert ^2&\ge \sum _{m\in \sigma } \Vert \varLambda _{m} h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _m h\Vert ^2\\&\ge \sum _{m\in \sigma } \frac{1}{B_{1,m}}\sum _{n\in J_m}|\langle \varLambda _{m} h,f_{n,m}\rangle |^2+\sum _{m\in \sigma ^c}\frac{1}{B_{2,m}}\sum _{n\in J_m}|\langle \varOmega _m h,g_{n,m}\rangle |^2\\&\ge \min \left\{ \frac{1}{B_1},\frac{1}{B_{2}}\right\} \left( \sum _{m\in \sigma }\sum _{n\in J_m}|\langle h,\varLambda _{m}^*f_{n,m}\rangle |^2+\sum _{m\in \sigma ^c}\sum _{n\in J_m}| h,\varOmega _m^*g_{n,m}\rangle |^2\right) . \end{aligned}$$

Therefore, $\{\varLambda _{m}^*f_{n,m}\}_{n\in J_m, m\in \mathbb {N}}$ and $\{\varOmega _{m}^*g_{n,m}\}_{n\in J_m, m\in \mathbb {N}}$ are weaving frames for $\mathcal {H}$.

To prove the converse part, suppose $\{\varLambda _{m}^*f_{n,m}\}_{n\in J_m, m\in \mathbb {N}}$ and $\{\varOmega _{m}^*g_{n,m}\}_{n\in J_m, m\in \mathbb {N}}$ are weaving frames for $\mathcal {H}$ with universal lower and upper frame bounds $\alpha$ and $\beta$, respectively.

Let $\sigma \subseteq \mathbb {N}$ and $h\in \mathcal {H}$ be arbitrary. Then,

$$\begin{aligned} \alpha \Vert h\Vert ^2&\le \sum _{m\in \sigma }\sum _{n\in J_m}|\langle h,\varLambda _{m}^*f_{n,m}\rangle |^2+\sum _{m\in \sigma ^c}\sum _{n\in J_m}|\langle h,\varOmega _m^*g_{n,m}\rangle |^2\\&=\sum _{m\in \sigma }\sum _{n\in J_m}|\langle \varLambda _{m}h,f_{n,m}\rangle |^2+\sum _{m\in \sigma ^c}\sum _{n\in J_m}|\langle \varOmega _m h,g_{n,m}\rangle |^2\\&\le \sum _{m\in \sigma }B_{1,m}\Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}B_{2,m}\Vert \varOmega _m h\Vert ^2\\&\le \sum _{m\in \sigma }B_1\Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}B_2\Vert \varOmega _m h\Vert ^2\\&\le \max \{B_1,B_2\}\left( \sum _{m\in \sigma }\Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _m h\Vert ^2\right) . \end{aligned}$$

Similarly,

$$\begin{aligned} \beta \Vert h\Vert ^2&\ge \sum _{m\in \sigma }\sum _{n\in J_m}|\langle h,\varLambda _{m}^*f_{n,m}\rangle |^2+\sum _{m\in \sigma ^c}\sum _{n\in J_m}|\langle h,\varOmega _m^*g_{n,m}\rangle |^2\\&=\sum _{m\in \sigma }\sum _{n\in J_m}|\langle \varLambda _{m}h,f_{n,m}\rangle |^2+\sum _{m\in \sigma ^c}\sum _{n\in J_m}|\langle \varOmega _m h,g_{n,m}\rangle |^2\\&\ge \min \{A_1,A_2\}\left( \sum _{m\in \sigma }\Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _m h\Vert ^2\right) . \end{aligned}$$

Therefore, $\{\varLambda _{m}\}_{ m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{ m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$. $\square$

Next example illustrates the above theorem.

Example 3.1

Suppose $\mathcal {H}$ is a separable Hilbert space with orthonormal basis $\{e_{n}\}_{n\in \mathbb {N}}$. For $m\in \mathbb {N}$, let $\mathcal {H}_m=\text {span}\{e_m,e_{m+1},e_{m+2}\}$ and define $\varLambda _{m}\in \mathcal {L}(\mathcal {H}, \mathcal {H}_m)$ by

$$\begin{aligned} \varLambda _{m}(h)&=\langle h,e_m\rangle e_m. \end{aligned}$$

Here, $\varLambda _{m}$ is the orthogonal projection of $\mathcal {H}$ onto $\text {span}\{e_m\}$, so $\varLambda _{m}^*=\varLambda _{m}$, and $\{2e_{m},2e_{m+1},2e_{m+2}\}$ is a frame for $\mathcal {H}_m$ with lower and upper frame bounds both equal to 4. For any $h\in \mathcal {H}$, we have

$$\begin{aligned} \sum _{m\in \mathbb {N}}\Vert \varLambda _{m}h\Vert ^2= \sum _{m\in \mathbb {N}}\Vert \langle h,e_m\rangle e_m\Vert ^2=\sum _{m\in \mathbb {N}}|\langle h,e_m\rangle |^2=\Vert h\Vert ^2. \end{aligned}$$

Therefore, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ is a g-frame for $\mathcal {H}$ w.r.t $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$.

I. For $m\in \mathbb {N}$, define $\varOmega _m\in \mathcal {L}(\mathcal {H}, \mathcal {H}_m)$ by

$$\begin{aligned} \varOmega _{m}(h)=\left\{ \begin{array}{ll}\langle h,e_1\rangle e_1+\langle h,e_2\rangle e_2, &{}\quad \text { if } m=1\\ \langle h,e_{m+1}\rangle e_{m+1}, &{}\quad \text { if } m\ge 2. \end{array}\right. \end{aligned}$$

Since $\varOmega _{m}$ is an orthogonal projection, so $\varOmega _{m}^*=\varOmega _{m}$. Also $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ is a g-frame for $\mathcal {H}$ w.r.t $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$.

For $\sigma =\{1\}$ and $h=e_2$, we compute

$$\begin{aligned}&\sum _{m\in \sigma }(|\langle h,\varLambda _m^*2e_m\rangle |^2+|\langle h,\varLambda _m^*2e_{m+1}\rangle |^2+|\langle h,\varLambda _m^*2e_{m+2}\rangle |^2)\\&\qquad +\sum _{m\in \sigma ^c}\left( |\langle h,\varOmega _m^*2e_m\rangle |^2+|\langle h,\varOmega _m^*2e_{m+1}\rangle |^2+|\langle h,\varOmega _m^*2e_{m+2}\rangle |^2\right) \\&\quad =\left( |\langle h,2\varLambda _1e_1\rangle |^2+|\langle h,2\varLambda _1e_{2}\rangle |^2+|\langle h,2\varLambda _1e_{3}\rangle |^2\right) \\&\qquad +\sum _{m=2}^\infty \left( |\langle h,2\varOmega _me_m\rangle |^2+|\langle h,2\varOmega _me_{m+1}\rangle |^2+|\langle h,2\varOmega _me_{m+2}\rangle |^2\right) \\&\quad =|\langle e_2,2e_1\rangle |^2+\sum _{m=2}^\infty |\langle e_2,2e_{m+1}\rangle |^2\\&\quad =0. \end{aligned}$$

Thus, $\{\varLambda _{m}^*2e_{m},\varLambda _{m}^*2e_{m+1},\varLambda _{m}^*2e_{m+2}\}_{m\in \sigma }\cup \{\varOmega _{m}^*2e_{m},\varOmega _{m}^*2e_{m+1},\varOmega _{m}^*2e_{m+2}\}_{m\in \sigma ^c}$ is not a frame for $\mathcal {H}$. Hence, $\{\varLambda _{m}^*2e_{m},\varLambda _{m}^*2e_{m+1},\varLambda _{m}^*2e_{m+2}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}^*2e_{m},\varOmega _{m}^*2e_{m+1},\varOmega _{m}^*2e_{m+2}\}_{m\in \mathbb {N}}$ are not weaving frames for $\mathcal {H}$, so by Theorem 3.1, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are not weaving g-frames for $\mathcal {H}$ .

II. For $m\in \mathbb {N}$, define $\varOmega _m\in \mathcal {L}(\mathcal {H}, \mathcal {H}_m)$ by

$$\begin{aligned} \varOmega _{m}(h)= \langle h,e_{m}\rangle e_{m}+\langle h,e_{m+1}\rangle e_{m+1}. \end{aligned}$$

Here $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ is a g-frame for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$ and $\varOmega _{m}^*=\varOmega _{m}$.

For $\sigma \subseteq \mathbb {N}$ and $h\in \mathcal {H}$, we have

$$\begin{aligned}&\sum _{m\in \sigma }(|\langle h,\varLambda _m^*2e_m\rangle |^2+|\langle h,\varLambda _m^*2e_{m+1}\rangle |^2+|\langle h,\varLambda _m^*2e_{m+2}\rangle |^2)\\&\qquad +\sum _{m\in \sigma ^c}\left( |\langle h,\varOmega _m^*2e_m\rangle |^2+|\langle h,\varOmega _m^*2e_{m+1}\rangle |^2+|\langle h,\varOmega _m^*2e_{m+2}\rangle |^2\right) \\&\quad =\sum _{m\in \sigma }|\langle h,2e_m\rangle |^2 +\sum _{m\in \sigma ^c}(|\langle h,2e_m\rangle |^2+|\langle h,2e_{m+1}\rangle |^2)\\&\quad =4\Vert h\Vert ^2+4\sum _{m\in \sigma ^c}|\langle h,e_{m+1}\rangle |^2 \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert h\Vert ^2&\le \sum _{m\in \sigma }(|\langle h,\varLambda _m^*2e_m\rangle |^2+|\langle h,\varLambda _m^*2e_{m+1}\rangle |^2+|\langle h,\varLambda _m^*2e_{m+2}\rangle |^2)\\&\quad +\sum _{m\in \sigma ^c}\left( |\langle h,\varOmega _m^*2e_m\rangle |^2+|\langle h,\varOmega _m^*2e_{m+1}\rangle |^2+|\langle h,\varOmega _m^*2e_{m+2}\rangle |^2\right) \le 8\Vert h\Vert ^2. \end{aligned}$$

Hence, $\{\varLambda _{m}^*2e_{m},\varLambda _{m}^*2e_{m+1},\varLambda _{m}^*2e_{m+2}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}^*2e_{m},\varOmega _{m}^*2e_{m+1},\varOmega _{m}^*2e_{m+2}\}_{m\in \mathbb {N}}$ are weaving frames for $\mathcal {H}$, so by Theorem 3.1, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ .

In the following theorem, we show the relation between optimal g-frame bounds and optimal universal g-frame bounds.

Theorem 3.2

Let $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ be g-frames for $\mathcal {H}$ w.r.t. $\{ \mathcal {H}_m\}_{m\in \mathbb {N}}$ with optimal lower g-frame bounds $A_1$, $A_2$ (respectively) and optimal upper g-frame bounds $B_1$, $B_2$ (respectively). If $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ with optimal universal lower and upper g-frame bounds A and B, respectively, then $A\le \min \{A_1,A_2\}$ and $B\ge \max \{B_1,B_2\}$.

Proof

Choose $\sigma =\mathbb {N}$. Then, for any $h\in \mathcal {H}$, we have

$$\begin{aligned} A\Vert h\Vert ^2\le \sum _{m\in \sigma } \Vert \varLambda _{m} h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _m h\Vert ^2=\sum _{m\in \mathbb {N}} \Vert \varLambda _{m} h\Vert ^2\le B\Vert h\Vert ^2. \end{aligned}$$

Thus, A and B are lower and upper g-frame bounds, respectively, of $\{\varLambda _{m}\}_{m\in \mathbb {N}}$. Since $A_1$ and $B_1$ are optimal g-frame bounds of $\{\varLambda _{m}\}_{m\in \mathbb {N}}$, $A\le A_1$ and $B\ge B_1$.

Similarly, $A\le A_2$ and $B\ge B_2$. Hence, $A\le \min \{A_1,A_2\}$ and $B\ge \max \{B_1,B_2\}$. $\square$

Let us illustrate an example of Theorem 3.2 where strict inequalities follow.

Example 3.2

Suppose $\mathcal {H}$ is a separable Hilbert space with orthonormal basis $\{e_{n}\}_{n\in \mathbb {N}}$. Let $\mathcal {H}_1=\text {span}\{e_1\}$, $\mathcal {H}_2=\mathcal {H}_3=\text {span}\{e_2\}$, $\mathcal {H}_4=\mathcal {H}_5=\text {span}\{e_3\}$ and for $m\in \mathbb {N}{\setminus }\{1,2,3,4,5\}$, let $\mathcal {H}_m=\text {span}\{e_{m-2}\}$.

For $m\in \mathbb {N}{\setminus }\{2,3,4,5\}$, $\varLambda _{m}$ and $\varOmega _{m}$ are orthogonal projection of $\mathcal {H}$ onto $\mathcal {H}_m$. Define $\varLambda _{2},\varLambda _{3},\varLambda _{4}, \varLambda _{5}, \varOmega _{2}, \varOmega _{3}, \varOmega _{4}, \varOmega _{5}$ as

$$\begin{aligned} \varLambda _{2}(h)&=\langle h,\frac{e_2}{\sqrt{2}}\rangle e_2,\ \varLambda _{3}(h)=\langle h,\frac{e_2}{\sqrt{2}}\rangle e_2,\ \varLambda _{4}(h)=\langle h,e_3\rangle e_3,\ \varLambda _{5}(h)=0\\ \varOmega _{2}(h)&=\langle h,e_2\rangle e_2,\ \varOmega _{3}(h)=0,\ \varOmega _{4}(h)=\langle h,\frac{e_3}{\sqrt{2}}\rangle e_3,\ \varOmega _{5}(h)=\langle h,\frac{e_3}{\sqrt{2}}\rangle e_3. \end{aligned}$$

For any $h\in \mathcal {H}$, we have

$$\begin{aligned} \sum _{m\in \mathbb {N}}\Vert \varLambda _{m}h\Vert ^2=\sum _{m\in \mathbb {N}}|\langle h,e_m\rangle |^2=\sum _{m\in \mathbb {N}}\Vert \varOmega _{m}h\Vert ^2=\Vert h\Vert ^2. \end{aligned}$$

Therefore, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are g-frames for $\mathcal {H}$ w.r.t $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$ with optimal lower and upper g-frame bounds equal to 1. Thus, $A_1=A_2=B_1=B_2=1$. Further for any $h\in \mathcal {H}$, we have

$$\begin{aligned} \frac{1}{2}|\langle h,e_{1}\rangle |^2&\le \Vert \varLambda _1h\Vert ^2=\Vert \varOmega _1h\Vert ^2\le \frac{3}{2}|\langle h,e_{1}\rangle |^2\\ \frac{1}{2}|\langle h,e_{2}\rangle |^2=\left| \left\langle h,\frac{e_{2}}{\sqrt{2}}\right\rangle \right| ^2&\le \Vert A_1\Vert ^2\le |\langle h,e_{2}\rangle |^2, \text { for }A_1=\varLambda _{2}h, \varOmega _{2}h\\ 0&\le \Vert A_2\Vert ^2\le \frac{1}{2}|\langle h,e_{2}\rangle |^2, \text { for }A_2=\varLambda _{3}h, \varOmega _{3}h\\ \frac{1}{2}|\langle h,e_{3}\rangle |^2&\le \Vert A_3\Vert ^2\le |\langle h,e_{3}\rangle |^2, \text { for }A_3=\varLambda _{4}h, \varOmega _{4}h\\ 0&\le \Vert A_4\Vert ^2\le \frac{1}{2}|\langle h,e_{3}\rangle |^2, \text { for }A_4=\varLambda _{5}h, \varOmega _{5}h\\ \frac{1}{2}|\langle h,e_{m-2}\rangle |^2&\le \Vert \varLambda _mh\Vert ^2=\Vert \varOmega _mh\Vert ^2\le \frac{3}{2}|\langle h,e_{m-2}\rangle |^2, \text { for all } m\ge 6. \end{aligned}$$

Let $\sigma \subseteq \mathbb {N}$ be arbitrary. Then, by using the above inequalities, we have

$$\begin{aligned} \frac{1}{2}\Vert h\Vert ^2&=\frac{1}{2}\sum _{m\in \mathbb {N}}|\langle h,e_m\rangle |^2\\&\le \sum _{m\in \sigma }\Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _{m}h\Vert ^2\\&\le \frac{3}{2}\sum _{m\in \mathbb {N}}|\langle h,e_m\rangle |^2=\frac{3}{2}\Vert h\Vert ^2. \end{aligned}$$

Thus, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ with universal lower and upper g-frame bounds $\frac{1}{2}$ and $\frac{3}{2}$, respectively.

For $\sigma _1=\{2\}$, $\sigma _2=\{4\}$, $h_1=e_2$, $h_2=e_3$, we have

$$\begin{aligned} \frac{1}{2}\Vert h_1\Vert ^2&=\sum _{m\in \sigma _1}\Vert \varLambda _{m}h_1\Vert ^2+\sum _{m\in \sigma _1^c}\Vert \varOmega _{m}h_1\Vert ^2\\ \frac{3}{2}\Vert h_2\Vert ^2&=\sum _{m\in \sigma _2}\Vert \varLambda _{m}h_2\Vert ^2+\sum _{m\in \sigma _2^c}\Vert \varOmega _{m}h_2\Vert ^2. \end{aligned}$$

Therefore, optimal universal upper g-frame bound is $\frac{3}{2}$ and optimal universal lower g-frame bound is $\frac{1}{2}$. Hence, $A=\frac{1}{2}<\min \{A_1,A_2\}$ and $B=\frac{3}{2}>\max \{B_1,B_2\}$.

In the next theorem, we show that the sum of the optimal g-frame bounds of two weaving g-frames is never the optimal universal g-frame bounds.

Theorem 3.3

Let $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ be g-frames for $\mathcal {H}$ w.r.t. $\{ \mathcal {H}_m\}_{m\in \mathbb {N}}$ with optimal lower g-frame bounds $A_1$, $A_2$ (respectively) and optimal upper g-frame bounds $B_1$, $B_2$ (respectively). If $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$, then $A_1+A_2$ is not the optimal universal lower g-frame bound and $B_1+B_2$ is not the optimal universal upper g-frame bound.

Proof

Suppose $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ with universal lower and upper g-frame bounds A and B, respectively. Since $\min \{A_1,A_2\}<A_1+A_2$, by Theorem 3.2, optimal universal lower g-frame bound is not equal to $A_1+A_2$.

Suppose that $B_1+B_2$ is the optimal universal upper g-frame bound of weaving g-frames $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$. For any $\epsilon >0$, $B_1+B_2-\epsilon$ is not a universal upper g-frame bound. Thus, there exists $\sigma \subseteq \mathbb {N}$ and $h\in \mathcal {H}$ such that

$$\begin{aligned} \sum \limits _{m\in \sigma } \Vert \varLambda _{m} h\Vert ^2+\sum \limits _{m\in \sigma ^c}\Vert \varOmega _m h\Vert ^2>(B_1+B_2-\epsilon )\Vert h\Vert ^2. \end{aligned}$$

Take $h_1=\frac{h}{\Vert h\Vert }$. Then,

$$\begin{aligned} \sum \limits _{m\in \sigma } \Vert \varLambda _{m} h_1\Vert ^2+\sum \limits _{m\in \sigma ^c}\Vert \varOmega _m h_1\Vert ^2>B_1+B_2-\epsilon \ge \sum \limits _{m\in \mathbb {N}} \Vert \varLambda _{m} h_1\Vert ^2+\sum \limits _{m\in \mathbb {N}}\Vert \varOmega _m h_1\Vert ^2-\epsilon \end{aligned}$$

Thus, we have

$$\begin{aligned} A\le \sum \limits _{m\in \sigma ^c} \Vert \varLambda _{m} h_1\Vert ^2+\sum \limits _{m\in \sigma }\Vert \varOmega _m h_1\Vert ^2<\epsilon . \end{aligned}$$

Since $\epsilon >0$ was arbitrary, so $A=0$, a contradiction. Therefore, $B_1+B_2$ is not the optimal universal upper g-frame bound. $\square$

Dual g-frames provide the series representation of each vector in $\mathcal {H}$ in terms of g-frame elements. Following theorem shows that a g-frame and its dual g-frame are woven.

Theorem 3.4

Let $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ be a g-frame for $\mathcal {H}$ w.r.t. $\{ \mathcal {H}_m\}_{m\in \mathbb {N}}$ with upper g-frame bound $B_1$ and let $\{\varGamma _{m}\}_{m\in \mathbb {N}}$ be its dual g-frame with upper g-frame bound $B_2$. Then, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varGamma _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ with universal lower g-frame bound $\min \left\{ \frac{1}{2B_1},\frac{1}{2B_2}\right\}$ and universal upper g-frame bound $B_1+B_2$.

Proof

Let $\sigma$ be any subset of $\mathbb {N}$. By using Cauchy–Schwartz inequality, we compute

$$\begin{aligned} \Vert h\Vert ^4&=|\langle h,h\rangle |^2\\&=\left| \left\langle \sum _{m\in \mathbb {N}}\varLambda _{m}^* \varGamma _m h,h\right\rangle \right| ^2\\&=\left| \sum _{m\in \sigma }\langle \varLambda _{m}^* \varGamma _m h,h\rangle +\sum _{m\in \sigma ^c}\langle \varLambda _{m}^* \varGamma _m h,h\rangle \right| ^2\\&=\left| \sum _{m\in \sigma }\langle \varGamma _m h,\varLambda _{m}h\rangle +\sum _{m\in \sigma ^c}\langle \varGamma _m h,\varLambda _{m}h\rangle \right| ^2\\&\le 2\left| \sum _{m\in \sigma }\langle \varGamma _m h,\varLambda _{m}h\rangle \right| ^2+2\left| \sum _{m\in \sigma ^c}\langle \varGamma _m h,\varLambda _{m}h\rangle \right| ^2\\ \quad \quad \quad&\le 2\sum _{m\in \sigma }\Vert \varGamma _m h\Vert ^2\sum _{m\in \sigma } \Vert \varLambda _{m}h\Vert ^2+2\sum _{m\in \sigma ^c}\Vert \varGamma _m h\Vert ^2\sum _{m\in \sigma ^c} \Vert \varLambda _{m}h\Vert ^2\\&\le 2B_2\Vert h\Vert ^2\sum _{m\in \sigma } \Vert \varLambda _{m}h\Vert ^2+2B_1\Vert h\Vert ^2\sum _{m\in \sigma ^c}\Vert \varGamma _m h\Vert ^2\\&\le \max \{2B_1,2B_2\}\Vert h\Vert ^2\left( \sum _{m\in \sigma } \Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varGamma _m h\Vert ^2\right) , \ \text {for all } h\in \mathcal {H}. \end{aligned}$$

Therefore,

$$\begin{aligned} \min \left\{ \frac{1}{2B_1},\frac{1}{2B_2}\right\}&\Vert h\Vert ^2\le \sum \limits _{m\in \sigma } \Vert \varLambda _{m}h\Vert ^2+\sum \limits _{m\in \sigma ^c}\Vert \varGamma _m h\Vert ^2\le (B_1+B_2)\Vert h\Vert ^2,\ \text {for all } h\in \mathcal {H}. \end{aligned}$$

$\square$

Since the g-frame operator S and its inverse are self-adjoint and positive, so their square roots exist. In the next theorem, we construct a new family of weaving g-frames using the existing family of weaving g-frames and the square root of $S^{-1}$.

Theorem 3.5

Let $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ be weaving g-frames for $\mathcal {H}$ w.r.t. $\mathcal {H}_m$. If S is the g-frame operator of $\{\varLambda _{m}\}_{m\in \mathbb {N}}$, then $\{\varLambda _{m}S^{-\frac{1}{2}}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}S^{-\frac{1}{2}}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ w.r.t. $\mathcal {H}_m$.

Proof

Suppose $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ with universal lower g-frame bound A and universal upper g-frame bound B. Then, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ is a g-frame for $\mathcal {H}$ with lower g-frame bound A and upper g-frame bound B, and hence $B^{-1}I\le S^{-1}\le A^{-1}I$, where I is the identity operator on $\mathcal {H}$.

For any subset $\sigma$ of $\mathbb {N}$ and $h\in \mathcal {H}$, we compute

$$\begin{aligned} \sum \limits _{m\in \sigma } \Vert \varLambda _{m}S^{-\frac{1}{2}}h\Vert ^2+\sum \limits _{m\in \sigma ^c}\Vert \varOmega _mS^{-\frac{1}{2}} h\Vert ^2&\ge A\Vert S^{-\frac{1}{2}}h\Vert ^2\\&= A\langle S^{-\frac{1}{2}}h,S^{-\frac{1}{2}}h\rangle \\&= A\langle (S^{-\frac{1}{2}})^*S^{-\frac{1}{2}}h,h\rangle \\&= A\langle S^{-\frac{1}{2}}S^{-\frac{1}{2}}h,h\rangle \\&= A\langle S^{-1}h,h\rangle \\&\ge \frac{A}{B}\Vert h\Vert ^2. \end{aligned}$$

For the universal upper g-frame bound, we have

$$\begin{aligned} \sum \limits _{m\in \sigma } \Vert \varLambda _{m}S^{-\frac{1}{2}}h\Vert ^2+\sum \limits _{m\in \sigma ^c}\Vert \varOmega _mS^{-\frac{1}{2}} h\Vert ^2&\le B\Vert S^{-\frac{1}{2}}h\Vert ^2\\&= B\langle S^{-\frac{1}{2}}h,S^{-\frac{1}{2}}h\rangle \\&= B\langle S^{-1}h,h\rangle \\&\le \frac{B}{A}\Vert h\Vert ^2. \end{aligned}$$

Therefore, $\{\varLambda _{m}S^{-\frac{1}{2}}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}S^{-\frac{1}{2}}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$ with universal lower g-frame bound $\frac{A}{B}$ and universal upper g-frame bound $\frac{B}{A}$. $\square$

4 Weaving Generalized Riesz Bases

We start this section with the definition of weaving g-Riesz bases and weaving g-orthonormal bases.

Definition 4.1

Two g-Riesz bases $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_{m} \}_{m\in \mathbb {N}}$ are called weaving g-Riesz bases if there exist positive constants $A\le B$ such that for any $\sigma \subseteq \mathbb {N}$, $\{\varLambda _{m}\}_{m\in \sigma }\cup \{\varOmega _{m}\}_{m\in \sigma ^c}$ is a g-Riesz basis for $\mathcal {H}$ with lower g-Riesz bound A and upper g-Riesz bound B.

The constants A and B are called universal lower g-Riesz bound and universal upper g-Riesz bound, respectively.

Definition 4.2

Two g-orthonormal bases $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_{m} \}_{m\in \mathbb {N}}$ are called weaving g-orthonormal bases if for any $\sigma \subseteq \mathbb {N}$, $\{\varLambda _{m}\}_{m\in \sigma }\cup \{\varOmega _{m}\}_{m\in \sigma ^c}$ is a g-orthonormal basis for $\mathcal {H}$.

In the next theorem, we show that weaving g-orthonormal bases remain to be woven even after applying unitary operator.

Theorem 4.1

If $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-orthonormal bases for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_{m}\}_{m\in \mathbb {N}}$ and U is any unitary operator on $\mathcal {H}$, then $\{\varLambda _{m}U\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}U\}_{m\in \mathbb {N}}$ are weaving g-orthonormal bases for $\mathcal {H}$.

Proof

Suppose $\sigma$ is any subset of $\mathbb {N}$. Then,

$$\begin{aligned} \sum \limits _{m\in \sigma } \Vert \varLambda _{m} Uh\Vert ^2+\sum \limits _{m\in \sigma ^c}\Vert \varOmega _mU h\Vert ^2=\Vert Uh\Vert ^2=\Vert h\Vert ^2,\ \text {for all } h\in \mathcal {H}. \end{aligned}$$

Suppose $m_1, m_2\in \mathbb {N}$ are arbitrary. Then, for any $h_{m_1}\in \mathcal {H}_{m_1}$ and $h_{m_2}\in \mathcal {H}_{m_2}$, we have

$$\begin{aligned}&\langle (\varLambda _{m_1}U)^*h_{m_1},(\varOmega _{m_2}U)^*h_{m_2}\rangle =\langle U^*\varLambda _{m_1}^*h_{m_1},U^*\varOmega _{m_2}^*h_{m_2}\rangle \\&\quad =\langle \varLambda _{m_1}^*h_{m_1},\varOmega _{m_2}^*h_{m_2}\rangle =\delta _{m_1,m_2}\langle h_{m_1},h_{m_2}\rangle \,\,\,\text {if } m_1\in \sigma ,\ m_2\in \sigma ^c,\\&\langle (\varLambda _{m_1}U)^*h_{m_1},(\varLambda _{m_2}U)^*h_{m_2}\rangle =\langle U^*\varLambda _{m_1}^*h_{m_1},U^*\varLambda _{m_2}^*h_{m_2}\rangle \\&\quad =\langle \varLambda _{m_1}^*h_{m_1},\varLambda _{m_2}^*h_{m_2}\rangle =\delta _{m_1,m_2}\langle h_{m_1},h_{m_2}\rangle \,\,\,\text {if } m_1,\ m_2 \in \sigma ,\\&\langle (\varOmega _{m_1}U)^*h_{m_1},(\varOmega _{m_2}U)^*h_{m_2}\rangle =\langle U^*\varOmega _{m_1}^*h_{m_1},U^*\varOmega _{m_2}^*h_{m_2}\rangle \\&\quad =\langle \varOmega _{m_1}^*h_{m_1},\varOmega _{m_2}^*h_{m_2}\rangle =\delta _{m_1,m_2}\langle h_{m_1},h_{m_2}\rangle \,\,\,\text {if } m_1, \ m_2\in \sigma ^c. \end{aligned}$$

Therefore, $\{\varLambda _{m}U\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}U\}_{m\in \mathbb {N}}$ are weaving g-orthonormal bases for $\mathcal {H}$. $\square$

We obtain the following corollary to the above theorem for g-orthonormal basis.

Corollary 4.1

If $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ is a g-orthonormal basis for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_{m}\}_{m\in \mathbb {N}}$ and U is any unitary operator on $\mathcal {H}$, then $\{\varLambda _{m}U\}_{m\in \mathbb {N}}$ is a g-orthonormal basis for $\mathcal {H}$.

Unitary operators are surjective isometries. Both surjectivity and isometry are necessary in Theorem 4.1 and Corollary 4.1 is justified in the following example.

Example 4.1

Suppose $\mathcal {H}=\ell ^2(\mathbb {N})$ with canonical orthonormal basis $\{e_{n}\}_{n\in \mathbb {N}}$. For $m\in \mathbb {N}$, let $\mathcal {H}_m=\text {span}\{e_m\}$ and define $\varLambda _{m}\in \mathcal {L}(\mathcal {H}, \mathcal {H}_m)$ by

$$\begin{aligned} \varLambda _{m}(h)=\langle h,e_m\rangle e_m. \end{aligned}$$

Here, $\varLambda _{m}$ is the orthogonal projection of $\mathcal {H}$ onto $\mathcal {H}_m$, so $\varLambda _{m}^*=\varLambda _{m}$.

For any $h\in \mathcal {H}$, we have

$$\begin{aligned} \sum _{m\in \mathbb {N}}\Vert \varLambda _{m}h\Vert ^2= \sum _{m\in \mathbb {N}}\Vert \langle h,e_m\rangle e_m\Vert ^2=\sum _{m\in \mathbb {N}}|\langle h,e_m\rangle |^2=\Vert h\Vert ^2. \end{aligned}$$

Suppose $m_1, m_2\in \mathbb {N}$ are arbitrary. Then, for any $h_{m_1}\in \mathcal {H}_{m_1}$ and $h_{m_2}\in \mathcal {H}_{m_2}$, we have

$$\begin{aligned} \langle \varLambda _{m_1}^*h_{m_1},\varLambda _{m_2}^*h_{m_2}\rangle&=\langle \varLambda _{m_1}h_{m_1},\varLambda _{m_2}h_{m_2}\rangle \\&=\langle \langle h_{m_1},e_{m_1}\rangle e_{m_1},\langle h_{m_2},e_{m_2}\rangle e_{m_2}\rangle \\&=\langle h_{m_1},e_{m_1}\rangle \langle e_{m _2}, h_{m_2}\rangle \langle e_{m_1}, e_{m_2}\rangle \\&=\langle h_{m_1},e_{m_1}\rangle \langle e_{m _2}, h_{m_2}\rangle \langle e_{m_1}, e_{m_2}\rangle \langle e_{m_1}, e_{m_2}\rangle \\&=\delta _{m_1,m_2}\langle h_{m_1},e_{m_1}\rangle \langle e_{m_2}, h_{m_2}\rangle \langle e_{m_1}, e_{m_2}\rangle ,\\ \langle h_{m_1}, h_{m_2}\rangle&=\langle \langle h_{m_1},e_{m_1}\rangle e_{m_1},\langle h_{m_2},e_{m_2}\rangle e_{m_2}\rangle \\&=\langle h_{m_1},e_{m_1}\rangle \langle e_{m_2}, h_{m_2}\rangle \langle e_{m_1}, e_{m_2}\rangle . \end{aligned}$$

Thus, $\langle \varLambda _{m_1}^*h_{m_1},\varLambda _{m_2}^*h_{m_2}\rangle =\delta _{m_1,m_2}\langle h_{m_1}, h_{m_2}\rangle$ and hence, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ is a g-orthonormal basis for $\mathcal {H}$ w.r.t. $\{\mathcal {H}_m\}_{m\in \mathbb {N}}$.

I. Define $U:\mathcal {H}\rightarrow \mathcal {H}$ by $U(h)=2h$. Then, U is a bounded, linear and surjective operator but U is not an isometry. For $h\in \mathcal {H}$, we have

$$\begin{aligned} \sum _{m\in \mathbb {N}}\Vert \varLambda _{m}Uh\Vert ^2=\sum _{m\in \mathbb {N}}\Vert \varLambda _{m}2h\Vert ^2= \sum _{m\in \mathbb {N}}\Vert \langle 2h,e_m\rangle e_m\Vert ^2=\sum _{m\in \mathbb {N}}|\langle 2h,e_m\rangle |^2=4\Vert h\Vert ^2. \end{aligned}$$

Therefore, $\{\varLambda _{m}U\}_{m\in \mathbb {N}}$ is not a g-orthonormal basis for $\mathcal {H}$. Hence, isometry of U is necessary in Corollary 4.1.

If $\varOmega _{m}=\varLambda _m$, $m\in \mathbb {N}$, then $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-orthonormal bases for $\mathcal {H}$. But $\{\varLambda _{m}U\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}U\}_{m\in \mathbb {N}}$ are not weaving g-orthonormal bases for $\mathcal {H}$ as $\{\varLambda _{m}U\}_{m\in \mathbb {N}}$ is not a g-orthonormal basis for $\mathcal {H}$. Hence, isometry of U is necessary in Theorem 4.1.

II. Define $U:\mathcal {H}\rightarrow \mathcal {H}$ by $U(a_1,a_2,a_3,a_4,\ldots )=(0,a_1,a_2,a_3,a_4,\ldots )$. Then, U is a bounded, linear and isometry operator but U is not surjective. Here U is a right shift operator, so $U^*$ is the left shift operator.

Since $\{e_m\}$ is an orthonormal basis of $\mathcal {H}_m$ and $\Vert (\varLambda _1U)^*e_1\Vert =\Vert U^*\varLambda _1e_1\Vert =\Vert U^*e_1\Vert =\Vert (0,0,0,\ldots )\Vert =0$, so $\{(\varLambda _m U)^*e_m\}_{m\in \mathbb {N}}$ is not an orthonormal basis for $\mathcal {H}$. By Theorem 2.3, $\{\varLambda _{m}U\}_{m\in \mathbb {N}}$ is not a g-orthonormal basis for $\mathcal {H}$. Hence, surjectivity of U is necessary in Corollary 4.1.

If $\varOmega _{m}=\varLambda _m$, $m\in \mathbb {N}$, then $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-orthonormal bases for $\mathcal {H}$. But $\{\varLambda _{m}U\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}U\}_{m\in \mathbb {N}}$ are not weaving g-orthonormal bases for $\mathcal {H}$. Hence, surjectivity of U is necessary in Theorem 4.1.

G-Riesz bases are generalization of Riesz bases but still some properties of weaving g-Riesz bases and weaving Riesz bases are different. Next two examples highlight these differences.

It is shown in Theorem 2.1 that if a frame and a Riesz basis are woven, then the frame must be a Riesz basis. However, this is not in the case of g-Riesz basis.

Remark 4.1

If a g-frame and a g-Riesz basis are woven then the g-frame need not be a g-Riesz basis.

Next example justifies the above remark.

Example 4.2

Suppose $\mathcal {H}$ is a separable Hilbert space with orthonormal basis $\{e_{n,m}\}_{n,m\in \mathbb {N}}$. For $m\in \mathbb {N}$, define $\varLambda _{m},\varOmega _{m}:\mathcal {H}\rightarrow \ell ^2(\mathbb {N})$ by

$$\begin{aligned}&\varLambda _{m}(h)={\left\{ \begin{array}{ll} \{\langle h,e_{n,k}\rangle \}_{k\in \mathbb {N}}\quad \text { if }m=2n\\ \{\langle h,e_{n,k}\rangle \}_{k\in \mathbb {N}}\quad \text { if }m=2n-1 \end{array}\right. }\\&\varOmega _{m}(h)={\left\{ \begin{array}{ll} \{\langle h,e_{n,2k}\rangle \}_{k\in \mathbb {N}}\quad \quad \text { if }m=2n\\ \{\langle h,e_{n,2k-1}\rangle \}_{k\in \mathbb {N}}\quad \text {if }m=2n-1. \end{array}\right. } \end{aligned}$$

For any $h\in \mathcal {H}$, we have

$$\begin{aligned} \sum _{m\in \mathbb {N}}\Vert \varLambda _{m}h\Vert ^2=2\sum _{n\in \mathbb {N}}\Vert \{\langle h,e_{n,k}\rangle \}_{k\in \mathbb {N}}\Vert ^2=2\sum _{n\in \mathbb {N}}\sum _{k\in \mathbb {N}}|\langle h,e_{n,k}\rangle |^2=2\Vert h\Vert ^2. \end{aligned}$$

Thus, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ is a g-frame for $\mathcal {H}$. Since $\{\varLambda _{m}\}_{m\in \mathbb {N}{\setminus }\{2\}}$ is also a g-frame for $\mathcal {H}$, so $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ is not a g-exact frame and hence it is not a g-Riesz basis for $\mathcal {H}$ as every g-Riesz basis is a g-exact frame.

Let $\{e_i\}_{i\in \mathbb {N}}$ be the canonical orthonormal basis for $\ell ^2(\mathbb {N})$. Then,

$$\begin{aligned} \varOmega _{m}^*(e_i)={\left\{ \begin{array}{ll} e_{n,2i-1}\quad &{}\text {if } m=2n-1\\ e_{n,2i} &{}\text {if } m=2n. \end{array}\right. } \end{aligned}$$

Since $\{\varOmega _{m}^*e_{i}\}_{i\in \mathbb {N},m\in \mathbb {N}}=\{e_{i,m}\}_{i\in \mathbb {N},m\in \mathbb {N}}$ is a Riesz basis for $\mathcal {H}$ being an orthonormal basis, so by Theorem 2.3, $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ is a g-Riesz basis for $\mathcal {H}$.

For any $\sigma \subseteq \mathbb {N}$ and $h\in \mathcal {H}$, we compute

$$\begin{aligned}&\sum _{m\in \sigma }\Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _{m}h\Vert ^2\nonumber \\&\quad =\sum _{m\in \sigma ,m=2n}\Vert \{\langle h,e_{n,k}\rangle \}_{k\in \mathbb {N}}\Vert ^2+\sum _{m\in \sigma ,m=2n-1}\Vert \{\langle h,e_{n,k}\rangle \}_{k\in \mathbb {N}}\Vert ^2\nonumber \\&\qquad +\sum _{m\in \sigma ^c,m=2n}\Vert \{\langle h,e_{n,2k}\rangle \}_{k\in \mathbb {N}}\Vert ^2+\sum _{m\in \sigma ^c,m=2n-1}\Vert \{\langle h,e_{n,2k-1}\rangle \}_{k\in \mathbb {N}}\Vert ^2\nonumber \\&\quad =\sum _{m\in \sigma ,m=2n}\sum _{k\in \mathbb {N}}|\langle h,e_{n,k}\rangle |^2+\sum _{m\in \sigma ,m=2n-1}\sum _{k\in \mathbb {N}}|\langle h,e_{n,k}\rangle |^2\nonumber \\&\qquad +\sum _{m\in \sigma ^c,m=2n}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k}\rangle |^2+\sum _{m\in \sigma ^c,m=2n-1}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k-1}\rangle |^2\nonumber \\&\quad =\sum _{n\in \mathbb {N}}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k}\rangle |^2+\sum _{n\in \mathbb {N}}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k-1}\rangle |^2\nonumber \\&\qquad +\sum _{m\in \sigma ,m=2n}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k-1}\rangle |^2+\sum _{m\in \sigma ,m=2n-1}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k}\rangle |^2\nonumber \\&\quad =\sum _{n\in \mathbb {N}}\sum _{k\in \mathbb {N}}|\langle h,e_{n,k}\rangle |^2+\sum _{m\in \sigma ,m=2n}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k-1}\rangle |^2+\sum _{m\in \sigma ,m=2n-1}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k}\rangle |^2\nonumber \\&\quad =\Vert h\Vert ^2+\sum _{m\in \sigma ,m=2n}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k-1}\rangle |^2+\sum _{m\in \sigma ,m=2n-1}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k}\rangle |^2\\&\quad \le \Vert h\Vert ^2+\sum _{n\in \mathbb {N}}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k-1}\rangle |^2+\sum _{n\in \mathbb {N}}\sum _{k\in \mathbb {N}}|\langle h,e_{n,2k}\rangle |^2\nonumber \\&\quad =\Vert h\Vert ^2+\sum _{n\in \mathbb {N}}\sum _{k\in \mathbb {N}}|\langle h,e_{n,k}\rangle |^2=2\Vert h\Vert ^2\nonumber . \end{aligned}$$

(3)

From (3), $\sum \nolimits _{m\in \sigma }\Vert \varLambda _{m}h\Vert ^2+\sum \nolimits _{m\in \sigma ^c}\Vert \varOmega _{m}h\Vert ^2\ge \Vert h\Vert ^2$. Hence, we have

$$\begin{aligned} \Vert h\Vert ^2\le \sum _{m\in \sigma }\Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _{m}h\Vert ^2\le 2\Vert h\Vert ^2. \end{aligned}$$

Therefore, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$, where $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ is a g-Riesz basis and $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ is not a g-Riesz basis.

It is presented in Theorem 2.2 that if two Riesz bases are woven, then every weaving is a Riesz basis. Since exact frames are same as Riesz bases, so conclusion of Theorem 2.2 also holds for exact frames, i.e., if two exact frames are woven, then every weaving is a Riesz basis (or an exact frame). But this is not true for g-exact frames.

Remark 4.2

If two g-exact frames $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$, then for $\sigma \subset \mathbb {N}$, $\{\varLambda _{m}\}_{m\in \sigma }\cup \{\varOmega _{m}\}_{m\in \sigma ^c}$ need not be a g-exact frame and hence need not be a g-Riesz basis for $\mathcal {H}$.

Next example justifies the above remark.

Example 4.3

Let $\mathcal {H}$ be a separable Hilbert space with orthonormal basis $\{e_n\}_{n\in \mathbb {N}}$. For $m\in \mathbb {N}$, define $\varLambda _{m},\varOmega _{m}:\mathcal {H}\rightarrow \mathbb {C}^4$ by

$$\begin{aligned} \varLambda _{m}(h)={\left\{ \begin{array}{ll}(\langle h,e_{2}\rangle ,\langle h,e_{4}\rangle ,\langle h,e_{1}\rangle ,0) \quad &{}\text { if }\, m=1\\ (\langle h,e_{2}\rangle ,\langle h,e_{4}\rangle ,\langle h,e_{3}\rangle ,0)\quad &{}\text { if }\, m=2\\ (\langle h,e_{2}\rangle ,\langle h,e_{4}\rangle ,\langle h,e_{5}\rangle ,\langle h,e_{6}\rangle )\quad &{}\text { if }\, m=3\\ (\langle h,e_{m+3}\rangle ,0,0,0)\quad &{}\text { if }\, m\ge 4\\ \end{array}\right. }\\ \varOmega _{m}(h)={\left\{ \begin{array}{ll}(\langle h,e_{1}\rangle ,\langle h,e_{3}\rangle ,\langle h,e_{2}\rangle ,0) &{}\text { if }\, m=1\\ (\langle h,e_{1}\rangle ,\langle h,e_{3}\rangle ,\langle h,e_{4}\rangle ,0) &{}\text { if }\, m=2\\ (\langle h,e_{1}\rangle ,\langle h,e_{3}\rangle ,\langle h,e_{5}\rangle ,\langle h,e_{6}\rangle )&{} \text { if }\, m=3\\ (\langle h,e_{m+3}\rangle ,0,0,0) &{}\text { if }\, m\ge 4. \end{array}\right. } \end{aligned}$$

Then, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are g-exact frames for $\mathcal {H}$. Further for any $h\in \mathcal {H}$, we have

$$\begin{aligned}&\sum _{i=1}^2|\langle h,e_{i}\rangle |^2\le \Vert A_1\Vert ^2\le \sum _{i=1}^4|\langle h,e_{i}\rangle |^2, \text { for }A_1=\varLambda _{1}h, \varOmega _{1}h\\&\sum _{i=3}^4|\langle h,e_{i}\rangle |^2\le \Vert A_2\Vert ^2\le \sum _{i=1}^4|\langle h,e_{i}\rangle |^2, \text { for }A_2=\varLambda _{2}h, \varOmega _{2}h\\&\sum _{i=5}^6|\langle h,e_{i}\rangle |^2\le \Vert A_3\Vert ^2\le \sum _{i=1}^6|\langle h,e_{i}\rangle |^2, \text { for }A_3=\varLambda _{3}h, \varOmega _{3}h\\&|\langle h,e_{m+3}\rangle |^2=\Vert \varLambda _{m}h\Vert ^2=\Vert \varOmega _{m}h\Vert ^2\le 3|\langle h,e_{m+3}\rangle |^2, \text { for all }m\ge 4. \end{aligned}$$

Let $\sigma \subseteq \mathbb {N}$ be arbitrary. Then, by using the above inequalities, we have

$$\begin{aligned} \Vert h\Vert ^2&=\sum _{m\in \mathbb {N}}|\langle h,e_{m}\rangle |^2\\&\le \sum _{m\in \sigma }\Vert \varLambda _{m}h\Vert ^2+\sum _{m\in \sigma ^c}\Vert \varOmega _{m}h\Vert ^2\\&\le 3\sum _{m\in \mathbb {N}}|\langle h,e_{m}\rangle |^2\\&=3\Vert h\Vert ^2. \end{aligned}$$

Thus, $\{\varLambda _{m}\}_{m\in \mathbb {N}}$ and $\{\varOmega _{m}\}_{m\in \mathbb {N}}$ are weaving g-frames for $\mathcal {H}$.

For $\sigma =\{1,2\}$, $\{\varLambda _{m}\}_{m\in \sigma }\cup \{\varOmega _{m}\}_{m\in \sigma ^c}$ is a g-frame for $\mathcal {H}$. Since $\{\varLambda _{m}\}_{m\in \sigma {\setminus }\{2\}}\cup \{\varOmega _{m}\}_{m\in \sigma ^c}$ is also a g-frame for $\mathcal {H}$, therefore, $\{\varLambda _{m}\}_{m\in \sigma }\cup \{\varOmega _{m}\}_{m\in \sigma ^c}$ is not a g-exact frame and hence not a g-Riesz basis for $\mathcal {H}$.