1 Introduction

The concept of frame was proposed by Duffin and Schaeffer [18] in 1952 to study nonharmonic Fourier series. It was reintroduced by Daubechies, Grossmann and Meyer [15] in 1986. Since then, the frame theory has developed greatly and attracted more attention and research. We refer to [12, 18, 23, 39] for basic results on frames. Recall that a countable sequence \(f=\{f_{j}\}_{j\in J}\) in a separable Hilbert space \({\mathcal {H}}\) is called a frame for \({\mathcal {H}}\) if there exist constants \(0<A\le B<\infty \) such that

$$\begin{aligned} A\Vert h\Vert ^{2}\le \sum _{j\in J}|\langle h,\,f_{j}\rangle |^{2}\le B\Vert h\Vert ^{2}{\hbox { }}(h\in {\mathcal {H}}). \end{aligned}$$
(1.1)

It is called a Bessel sequence with bound B if the right-hand side inequality of (1.1) holds. A frame for \({\mathcal {H}}\) is called a Riesz basis for \({\mathcal {H}}\) if it ceases to a frame when removing an arbitrary element. A sequence \(f=\{f_{j}\}_{j\in J}\) in \({\mathcal {H}}\) is called a frame sequence (Riesz sequence) if it is a frame (Riesz basis) for its closed linear span. For a Bessel sequence \(f=\{f_{j}\}_{j\in J}\) in \({\mathcal {H}}\), we always denote by \(T_{f}\), \(T_{f}^{*}\) and \(S_{f}\) its synthesis operator \(T_{f}:l^{2}\rightarrow {\mathcal {H}}\), analysis operator \(T_{f}^{*}:{\mathcal {H}}\rightarrow l^{2} \) and frame operator \(S_{f}: {\mathcal {H}}\rightarrow {\mathcal {H}}\), respectively, i.e.,

$$\begin{aligned}&T_{f}c=\sum _{j\in J}c_{j}f_{j}\,(c=\{c_{j}\}_{j\in J}\in l^{2}) ,\\&T_{f}^{*}h=\{\langle h,\,f_{j}\rangle \}_{j\in J}\,(h\in {\mathcal {H}}),\\&S_{f}h=T_{f}T_{f}^{*}h=\sum _{j\in J}\langle h,\,f_{j}\rangle f_{j}\,(h\in {\mathcal {H}}). \end{aligned}$$

In last decades, various generalizations of frame have been put forward for special purposes. The concept of Hilbert–Schmidt frame (HS-frame for simplicity) was introduced by Sadeghi and Arefijamaal in [34] which is more general than that of all existing frames such as g-frame, frame of subspaces, pseudo-frame, fusion frame, outer frame, bounded quasi-projector and a class of time-frequency localization operators (see [3, 5, 7, 8, 17, 20, 29, 35, 36]).

Given two Hilbert spaces \({\mathcal {H}}_{1}\) and \({\mathcal {H}}_{2}\), we denote by \({\mathcal {B}}({\mathcal {H}}_{1},\,{\mathcal {H}}_{2})\) the set of all bounded linear operators from \({\mathcal {H}}_{1}\) to \({\mathcal {H}}_{2}\), and write \({\mathcal {B}}({\mathcal {H}}_{1})={\mathcal {B}}({\mathcal {H}}_{1},\,{\mathcal {H}}_{1})\). For a sequence \(\{\Theta _{j}\}_{j\in J}\) in \({\mathcal {B}}({\mathcal {H}}_{1},\,{\mathcal {H}}_{2})\), \(\Theta :=\sum \limits _{j\in J}\Theta _{j}\) means that, for each \(h\in {\mathcal {H}}_{1}\), \(\Theta h=\sum \limits _{j\in J}\Theta _{j}h\) with unconditional convergence. Throughout this paper, \({\mathcal {H}}\) and \({\mathcal {K}}\) are separable Hilbert spaces, J and K are countable index sets. Given \(U\in {\mathcal {B}}({\mathcal {K}})\) and an orthonormal basis \(\{e_{k}\}_{k\in K}\) for \({\mathcal {K}}\), define

$$\begin{aligned} \Vert U\Vert _{2}=\left( \sum _{k\in K}\Vert Ue_{k}\Vert ^{2}\right) ^{\frac{1}{2}}. \end{aligned}$$

It is well known that, \(\Vert U\Vert _{2}\) is independent of the choice of \(\{e_{k}\}_{k\in K}\). We call the operator U a Hilbert–Schmidt operator if \(\Vert U\Vert _{2}<\infty \). Let \({\mathcal {C}}_{2}({\mathcal {K}})\) (\({\mathcal {C}}_{2}={\mathcal {C}}_{2}({\mathcal {K}})\) for simplicity) denote the Hilbert space consisting of all Hilbert–Schmidt operators on \({\mathcal {K}}\) endowed with the inner product

$$\begin{aligned} \langle U,V\rangle _{\tau }=\sum _{k\in K}\langle Ue_{k},\, Ve_{k}\rangle \,(U, V\in {\mathcal {C}}_{2}). \end{aligned}$$

Define \(\oplus {\mathcal {C}}_{2}\) as the Hilbert space:

$$\begin{aligned} \oplus {\mathcal {C}}_{2}=\left\{ {\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}{\hbox { and }}\Vert {\mathcal {A}}\Vert _{2}=\left( \sum _{j\in J}\big \Vert {\mathcal {A}}_{j}\big \Vert ^{2}_{2}\right) ^{\frac{1}{2}}<\infty \right\} \end{aligned}$$

with the inner product given by \(\langle {\mathcal {A}},{\mathcal {B}}\rangle =\sum \limits _{j\in J}\langle {\mathcal {A}}_{j},{\mathcal {B}}_{j}\rangle _{\tau }\) for \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\) and \({\mathcal {B}}=\{{\mathcal {B}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\). Write

$$\begin{aligned} \oplus _{0}{\mathcal {C}}_{2}=\left\{ {\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}:{\mathcal {A}}\in \oplus {\mathcal {C}}_{2}{\hbox { and }}{\mathcal {A}}_{j}=0{\hbox { for }}j\notin J_{0}{\hbox { with some finite subset }}J_{0}{\hbox { of }}J\right\} . \end{aligned}$$

Definition 1.1

( [16, 32, 34]) A sequence \(\{{\mathcal {G}}_{j}\}_{j\in J}\) in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) is said to be a Hilbert–Schmidt frame or simply a HS-frame for \({\mathcal {H}}\) (a HS-frame sequence in \({\mathcal {H}}\)) with respect to \({\mathcal {K}}\), if there exist constants \(0<A\le B<\infty \) such that

$$\begin{aligned} A\Vert h\Vert ^{2}\le \sum _{j\in J}\Big \Vert {\mathcal {G}}_{j}h\Big \Vert _{2}^{2}\le B\Vert h\Vert ^{2}\left( h\in {\mathcal {H}}\,(h\in \overline{\mathrm{span}}\{{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}\}_{j\in J})\right) , \end{aligned}$$
(1.2)

where A and B are called HS-frame bounds. It is called a Parseval HS-frame if \(A=B=1\) in (1.2). It is called a HS-Bessel sequence with bound B if the right-hand side inequality of (1.2) holds. It is called HS-complete if \(\{h\in {\mathcal {H}}: {\mathcal {G}}_{j}h=0{\hbox { for }} j\in J\}=\{0\}\). It is called a HS-Riesz sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) if there exist constants \(0<A\le B<\infty \) such that

$$\begin{aligned} A\Vert {\mathcal {A}}\Vert ^{2}\le \left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}\right\| ^{2}\le B\Vert {\mathcal {A}}\Vert ^{2}\,\left( {\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0}{\mathcal {C}}_{2}\right) , \end{aligned}$$
(1.3)

where A and B are called HS-Riesz bounds. It is called a HS-Riesz basis for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) if it is a HS-Riesz sequence and HS-complete. It is called a HS-orthonormal system in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) if \({\mathcal {G}}_{j}{\mathcal {G}}_{k}^{*}=\delta _{j,\,k}I\) for \(j,k\in J\). And it is called a HS-orthonormal basis for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) if it is a HS-orthonormal system and HS-complete. Herein and after, we use I to denote the identity operator regardless of its underlying space.

Given a HS-Bessel sequence \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\), we always denote by \(T_{{\mathcal {G}}}\), \(T_{{\mathcal {G}}}^{*}\) and \(S_{{\mathcal {G}}}\) its synthesis operator \(T_{{\mathcal {G}}}:\oplus {\mathcal {C}}_{2}\rightarrow {\mathcal {H}}\), analysis operator \(T_{{\mathcal {G}}}^{*}:{\mathcal {H}}\rightarrow \oplus {\mathcal {C}}_{2}\) and frame operator \(S_{{\mathcal {G}}}: {\mathcal {H}}\rightarrow {\mathcal {H}}\), respectively, i.e.,

$$\begin{aligned}&T_{{\mathcal {G}}}{\mathcal {A}}=\sum _{j\in J}{\mathcal {G}}^{*}_{j}{\mathcal {A}}_{j}\,\left( {\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\right) ,\\&T_{{\mathcal {G}}}^{*}h=\{{\mathcal {G}}_{j}h\}_{j\in J}\,\left( h\in {\mathcal {H}}\right) ,\\&S_{{\mathcal {G}}}h=T_{{\mathcal {G}}}T_{{\mathcal {G}}}^{*}h=\sum _{j\in J}{\mathcal {G}}^{*}_{j}{\mathcal {G}}_{j}h\,\left( h\in {\mathcal {H}}\right) . \end{aligned}$$

For more details on HS-frames, we refer the reader to [4, 16, 28, 30,31,32,33,34].

Given a matrix \(M=(m_{i,\,j})_{i,\,j\in \mathbb {N}}\) and two sequences \(\{f_{i}\}_{i=1}^{\infty }\), \(\{g_{i}\}_{i=1}^{\infty }\) in a Hilbert space \({\mathcal {H}}\), \((g_{1},g_{2},\cdots )=(f_{1},f_{2},\cdots )M\) means that, for each \(j\in \mathbb {N}\), \(\sum \limits _{i=1}^{\infty }m_{i,\,j}f_{i}\) is convergent and \(g_{j}=\sum \limits _{i=1}^{\infty }m_{i,\,j}f_{i}\). It is well known that permutating a frame for \({\mathcal {H}}\) leaves a frame for \({\mathcal {H}}\). And by a standard argument, given a frame for \({\mathcal {H}}\), its image under an arbitrary bounded linear bijection on \({\mathcal {H}}\) is still a frame for \({\mathcal {H}}\). But the latter excludes permutation. For example, if \(\{e_{1}, e_{2},\cdots \}\) is an orthonormal basis for \({\mathcal {H}}\), then

$$\begin{aligned} \{f_{i}\}_{i=1}^{\infty }=\{0,e_{1}, e_{2},\cdots \} \end{aligned}$$
(1.4)

and its permutation

$$\begin{aligned} \{g_{i}\}_{i=1}^{\infty }=\{e_{1},0, e_{2},\cdots \} \end{aligned}$$
(1.5)

are both Parseval frames for \({\mathcal {H}}\). However, there exists no \(T\in {\mathcal {B}}({\mathcal {H}})\) such that \(g_{i}=Tf_{i}\) for all \(i\in \mathbb {N}\). Indeed,

$$\begin{aligned} Tf_{1}=0\ne e_{1}=g_{1} \end{aligned}$$

for an arbitrary \(T\in {\mathcal {B}}({\mathcal {H}})\). Fortunately,

$$\begin{aligned} (e_{1},0,e_{2},\cdots )=(0,e_{1},e_{2},\cdots )M, \end{aligned}$$

where \(M= \begin{pmatrix} M_{1} &{} 0 \\ 0 &{} I \end{pmatrix}\), \(M_{1}= \begin{pmatrix} 0 &{} c \\ 1 &{} 0 \end{pmatrix}\), c is an arbitrary constant. Obviously, M defines a bounded linear operator on \(l^{2}\), and M is invertible if and only if \(c\ne 0\). More general, \(l^{2}\)-operator images of a frame contains its all permutated versions. Aldroubi in [2] studied \(l^{2}\)-operator portraits of frames. The author characterized all bounded linear operators on \(l^{2}\) that transform frames into other frames for \({\mathcal {H}}\). Specifically, given a frame \(\{f_{i}\}_{i=1}^{\infty }\) for \({\mathcal {H}}\), the author characterized all bounded linear operators \({\widetilde{M}}\) on \(l^{2}\) such that \(\{g_{i}\}_{i=1}^{\infty }\) defined below is a frame for \({\mathcal {H}}\):

$$\begin{aligned} (g_{1},g_{2},\cdots )=(f_{1},f_{2},\cdots ){\widetilde{M}}. \end{aligned}$$
(1.6)

This method was also applied to g-frame setting in Guo [22]. Observe that \({\widetilde{M}}\) in (1.6) is linear. We can think that \(\{g_{i}\}_{i=1}^{\infty }\) is a linear perturbation of \(\{f_{i}\}_{i=1}^{\infty }\). On the other hand, nonlinear perturbation (usually so-called perturbation) of frames was extensively studied ([6, 9,10,11, 13, 14, 19, 24, 25]). It can be considered as an extension of the classical Paley–Wiener Theorem in [39] which reads: Let \(\{x_{i}\}_{i=1}^{\infty }\) be a basis for a Banach space X, then an arbitrary sequence \(\{y_{i}\}_{i=1}^{\infty }\) in X satisfying the following condition is a basis for X: there exists a constant \(0\le \lambda <1\) such that

$$\begin{aligned} \left\| \sum _{i=1}^{\infty }c_{i}(x_{i}-y_{i})\right\| \le \lambda \left\| \sum _{i=1}^{\infty }c_{i}x_{i}\right\| \,(c\in l_{0}). \end{aligned}$$

Recall from [6, 9, 11, 14] that the Paley–Wiener type theorem on frames says that, given a frame (frame sequence) \(\{f_{i}\}_{i=1}^{\infty }\) in a Hilbert space \({\mathcal {H}}\), a sequence \(\{g_{i}\}_{i=1}^{\infty }\) in \({\mathcal {H}}\) is a frame (frame sequence) in \({\mathcal {H}}\) if there exist constants \(\lambda _{1},\,\lambda _{2},\,\mu \ge 0\) such that

$$\begin{aligned} \left\| \sum _{i=1}^{\infty }c_{i}(f_{i}-g_{i})\right\| \le \lambda _{1}\left\| \sum _{i=1}^{\infty }c_{i}f_{i}\right\| +\lambda _{2}\left\| \sum _{i=1}^{\infty }c_{i}g_{i}\right\| +\mu \left( \sum _{i=1}^{\infty }|c_{i}|^{2}\right) ^{\frac{1}{2}} \,(c\in l_{0}), \end{aligned}$$
(1.7)

where \(0\le \lambda _{2}<1\), and \(\lambda _{1}\), \(\mu \) satisfy some technical conditions. It was generalized to g-frame, K-g-frame and HS-frame settings ( [1, 21, 27, 32, 36, 38]). By [6, Theorem 2], if \(\{f_{i}\}_{i=1}^{\infty }\) is a frame with lower bound A and \(\max (\lambda _{1}+\frac{\mu }{\sqrt{A}},\,\lambda _{2})<1\) in (1.7), then \(\{g_{i}\}_{i=1}^{\infty }\) is a frame.

Suppose \(\{f_{i}\}_{i=1}^{\infty }\) and \(\{g_{i}\}_{i=1}^{\infty }\) are as in (1.4) and (1.5). Then \(\left\| \sum \limits _{i=1}^{\infty }c_{i}(f_{i}-g_{i})\right\| =1\) if \(c_{2}=1\) and \(c_{i}=0\) for \(i\ne 2\). Thus (1.7) cannot holds for arbitrary positive constants \(\lambda _{1}\), \(\lambda _{2}\) and \(\mu \) satisfying \(\max (\lambda _{1}+\mu ,\,\lambda _{2})<1\). This suggests to some extent that the perturbation technique cannot cover \(l^{2}\)-operator portrait method.

Motivated by the above works, this paper addresses \(l^{2}-\)operator portraits and perturbations under the setting of HS-frame sequences. It is well known that an orthonormal basis for a Hilbert space may represent its all elements. However, Example 2.1 below shows that a HS-orthonormal basis for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) need not represent all elements in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\), not to mention a HS-frame. So, to study frame properties of \(l^{2}\)-operator portraits of HS-frames (HS-frame sequences), we naturally restrict us to \(l^{2}\)-decomposable HS-operator sequences.

Definition 1.2

Given a HS-orthonormal basis \({\mathcal {E}}=\{{\mathcal {E}}_{j}\}_{j\in J}\) for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\), a sequence \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) is said to be \(l^{2}(J)\)-decomposable with respect to \({\mathcal {E}}\) is there exists \(U=(u_{k,\,j})_{k,\,j\in J}\in {\mathcal {B}}(l^{2}(J))\) such that

$$\begin{aligned} {\mathcal {G}}_{j}=\sum _{k\in J}u_{k,\,j}{\mathcal {E}}_{k}\,(j\in J). \end{aligned}$$

Remark 1.1

By Lemmas 2.2 and 2.4 below, \({\mathcal {G}}\) is well defined and is a HS-Bessel sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\).

From [11, 14, 37], the “gap between two subspaces” is important in studying perturbation of frame sequences. It is also useful in this paper. Let V, W be two closed subspaces of \({\mathcal {H}}\). When \(V\ne \{0\}\), the gap from V to W is defined by

$$\begin{aligned} \delta (V,\,W)=\sup _{x\in V,\Vert x\Vert =1}\mathrm{dist}(x,\,W)=\sup _{x\in V,\Vert x\Vert =1}\inf _{y\in W}\Vert x-y\Vert . \end{aligned}$$

Define \(\delta (V,\,W)=0\) when \(V=\{0\}\). From [26] we know that \(\delta (V,\,W)=\delta (W^{\perp },V^{\perp })\), \(0\le \delta (V,\,W)\le 1\), and that \(\delta (V,\,W)\) need not equal to \(\delta (W,\,V)\). Recall from [16, Lemma 2.5] and [32, Theorem 4.3] that a HS-Bessel sequence is a HS-frame sequence if and only if its synthesis operator has closed range, and that suitably perturbing a HS-frame \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) leaves a HS-frame \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\), where \(\mathrm{range}(T_{{\mathcal {F}}})\subset \mathrm{range}(T_{{\mathcal {G}}})\) automatically holds. But we need not have \(\mathrm{range}(T_{{\mathcal {F}}})\subset \mathrm{range}(T_{{\mathcal {G}}})\) if \({\mathcal {G}}\) is a HS-frame sequence but not a HS-frame. This suggests us to study stable perturbation of HS-frame sequences with the help of gap between subspaces associated with \(T_{{\mathcal {F}}}\) and \(T_{{\mathcal {G}}}\).

The rest of this paper is organized as follows. In Sect. 2, we prove that an arbitrary bounded invertible operator on \(l^{2}(J)\) transforms a HS-frame into another HS-frame (see Theorem 2.1); and present a sufficient condition on bounded operators on \(l^{2}(J)\) which transform an \(l^{2}(J)\)-decomposable HS-frame into another HS-frame (HS-Riesz basis, HS-frame sequence and HS-Riesz sequence) (see Theorem 2.2). In Sect. 3, we prove that suitably perturbing a HS-frame sequence (HS-Riesz sequence) leaves a HS-frame sequence (HS-Riesz sequence). In Sect. 4, we give a remark that shows our results can recover some known conclusions on frames.

Before proceeding, we introduce some notations and notions for later use. For a HS-frame \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\), another HS-frame \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) is called an alternate dual HS-frame of \({\mathcal {G}}\) if

$$\begin{aligned} h=\sum _{j\in J}{\mathcal {G}}^{*}_{j}{\mathcal {F}}_{j}h{\hbox { }}(h\in {\mathcal {H}}). \end{aligned}$$

In particular, \(\{{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}\}_{j\in J}\) is a dual of \({\mathcal {G}}\) by [34, Remark 3.8], which is called the canonical dual HS-frame of \({\mathcal {G}}\) in [4]. If \({\mathcal {G}}\) is a HS-frame sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\), by a standard argument, \(\overline{\mathrm{span}}\{({\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1})^{*}{\mathcal {A}}_{j}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}\}_{j\in J}=\overline{\mathrm{span}}\{{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}\}_{j\in J}\). Thus \(\{{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}\}_{j\in J}\) is also a HS-frame sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with HS-frame bounds \(B^{-1}\) and \(A^{-1}\). In this paper, \(P_{M}\) always denotes the orthogonal projection from an appropriate space to the closed subspace M. For a bounded linear operator U, we denote by \(\ker (U)\), \(\mathrm{{range}}(U)\) and \(U^{*}\) its kernel, range and adjoint operator, respectively, and by \(U^{\dagger }\) its pseudo-inverse if U is of closed range in addition. For a matrix U, we denote by \({\overline{U}}\) and \(U^{*}\) its conjugate and conjugate transpose, respectively.

2 \(l^{2}(J)\)-operator Portraits

This section focuses on \(l^{2}(J)\)-operator portraits of HS-frames (HS-frame sequences). We prove that “HS-frame sequence” property is preserved under bounded invertible operators on \(l^{2}(J)\), and present a sufficient condition on bounded operators on \(l^{2}(J)\) which transform an \(l^{2}(J)\)-decomposable HS-frame into another HS-frame (HS-Riesz basis, HS-frame sequence and HS-Riesz sequence). For this purpose, we need to give some lemmas. Before this, we first given an example that a HS-operator does not represented by a HS-orthonormal basis. This explains why we introduce Definition 1.2.

Example 2.1

Let \({\mathcal {H}}\) be a Hilbert space with \(\{e_{(i,\,j)}\}_{(i,\,j)\in \mathbb {N}^{2}}\) being an orthonormal basis for \({\mathcal {H}}\), and \({\mathcal {K}}=l^{2}(\mathbb {N})\) with \(\{w_{k}\}_{k\in \mathbb {N}}\) be the standard orthonormal basis that is, for every k, \(w_{k}\) is 1 in the \(k-th\) position and 0 in other positions. Define \(\{\xi _{(i,\,j)}\}_{(i,\,j)\in \mathbb {N}^{2}}\) by

$$\begin{aligned} \xi _{(i,\,j)}w_{k}=\delta _{k,\,j}Uw_{i} {\hbox { }}(k\in \mathbb {N}), \end{aligned}$$
(2.1)

where \(U=\left( \begin{array}{cc} U_{1} &{} 0 \\ 0 &{} I \end{array} \right) \), \(U_{1}=\left( \begin{array}{cc} \frac{1}{\sqrt{2}} &{} \frac{1}{\sqrt{2}} \\ \\ -\frac{1}{\sqrt{2}} &{} \frac{1}{\sqrt{2}} \end{array} \right) \). Then \(\{\xi _{(i,\,j)}\}_{(i,\,j)\in \mathbb {N}^{2}}\) is an orthonormal basis for \({\mathcal {C}}_{2}\) by [30, Theorem2.1], and define the operator sequence \(\{{\mathcal {G}}_{(i,\,j)}\}_{(i,\,j)\in \mathbb {N}^{2}}\) in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) by

$$\begin{aligned} {\mathcal {G}}_{(i,\,j)}e_{(l,m)}={\left\{ \begin{array}{ll} \xi _{(i,\,j)}\,&{}\text {if}\,\,\,(l,\,m)=(i,\,j);\\ \\ 0\,&{}otherwise. \end{array}\right. } \end{aligned}$$
(2.2)

Then \(\{{\mathcal {G}}_{(i,\,j)}\}_{(i,\,j)\in \mathbb {N}^{2}}\) is a Parseval HS-frame for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\). By [31, Theorem 2.1], there exists a Hilbert space \(\widetilde{{\mathcal {H}}}={\mathcal {H}}\bigoplus {\mathcal {M}}\supset {\mathcal {H}}\) and a HS-orthonormal basis \(\{\widetilde{{\mathcal {G}}}_{(i,\,j)}\}_{(i,\,j)\in \mathbb {N}^{2}}\) for \(\widetilde{{\mathcal {H}}}\) with respect to \({\mathcal {K}}\) such that

$$\begin{aligned} \widetilde{{\mathcal {G}}}_{(i,\,j)}|_{{\mathcal {H}}}={\mathcal {G}}_{(i,\,j)}{\hbox { }}\left( (i,\,j)\in \mathbb {N}^{2}\right) . \end{aligned}$$
(2.3)

Define \({\widetilde{\Lambda }}\in {\mathcal {B}}(\widetilde{{\mathcal {H}}},\,{\mathcal {C}}_{2})\) by

$$\begin{aligned} {\widetilde{\Lambda }}(h\oplus g)=\langle h,\,e_{(1,\,1)}\rangle {\widetilde{\xi }}{\hbox { }}\left( h\oplus g\in \widetilde{{\mathcal {H}}}\right) , \end{aligned}$$
(2.4)

where \({\widetilde{\xi }}\) is defined by

$$\begin{aligned} {\widetilde{\xi }}\, w_{k}=\delta _{1,\,k}w_{1}{\hbox { }}\left( k\in \mathbb {N}\right) . \end{aligned}$$
(2.5)

Then we claim that there exists no \(c=\{c_{(i,\,j)}\}_{(i,\,j)\in \mathbb {N}^{2}}\in l^{2}\) such that

$$\begin{aligned} {\widetilde{\Lambda }}=\sum _{(i,\,j)\in \mathbb {N}^{2}}c_{(i,\,j)}\widetilde{{\mathcal {G}}}_{(i,\,j)}. \end{aligned}$$
(2.6)

We prove it by contradiction. Suppose (2.6) holds for some \(c\in l^{2}\). Observe that

$$\begin{aligned} \widetilde{\mathcal {G}}_{(i,\,j)}(e_{(1,\,1)}\oplus 0)w_{1}={\mathcal {G}}_{(i,\,j)}e_{(1,\,1)}w_{1}=\delta _{(i,\,j),(1,\,1)}\left( \begin{array}{c} \frac{1}{\sqrt{2}} \\ \\ -\frac{1}{\sqrt{2}}\\ \\ 0 \\ \vdots \end{array} \right) \end{aligned}$$

by (2.1)–(2.3). It follows that

$$\begin{aligned} {\widetilde{\Lambda }}(e_{(1,\,1)}\oplus 0)w_{1}=c_{(1,\,1)}\left( \begin{array}{c} \frac{1}{\sqrt{2}} \\ \\ -\frac{1}{\sqrt{2}}\\ \\ 0 \\ \vdots \end{array} \right) . \end{aligned}$$
(2.7)

However, using (2.4) and (2.5) we have

$$\begin{aligned} {\widetilde{\Lambda }}(e_{(1,\,1)}\oplus 0)w_{1}=\left( \begin{array}{c} 1 \\ \\ 0 \\ \vdots \end{array} \right) . \end{aligned}$$

It contradicts (2.7).

The following lemma reveals the connection between the pseudo-inverse of the product of two matrices and their pseudo-inverses.

Lemma 2.1

Let U be an \(l\times n\) matrix and V be an \(n\times m\) matrix. Suppose \(\mathrm{{range}}(V)\subset \mathrm{{range}}(U^{*})\). Then \((UV)^{\dagger }x=V^{\dagger }U^{\dagger }x\) for \(x\in \mathrm{{range}}(UV)\). Furthermore, \(\Vert (UV)^{\dagger }\Vert \le \Vert U^{\dagger }\Vert \Vert V^{\dagger }\Vert \).

Proof

Arbitrarily fix \(x\in \mathrm{{range}}(UV)\). Then

$$\begin{aligned} z=(UV)^{\dagger }x\in \ker (UV)^{\perp }{\hbox { and }}UVz=x \end{aligned}$$
(2.8)

by the definition of pseudo-inverse. This implies that

$$\begin{aligned} z\in \ker (V)^{\perp } \end{aligned}$$
(2.9)

due to \(\ker (V)\subset \ker (UV)\). Since \(\mathrm{{range}}(V)\subset \mathrm{{range}}(U^{*})=\ker (U)^{\perp }\), we have

$$\begin{aligned} Vz\in \ker (U)^{\perp }. \end{aligned}$$
(2.10)

Observe that \(\mathrm{range}(U^{\dagger })=\ker (U)^{\perp }\) and \(U^{\dagger }U=P_{\mathrm{range}(U^{\dagger })}\) by [12, Lemma 2.5.2]. It follows that \(U^{\dagger }U=P_{\ker (U)^{\perp }}\). Similarly, \(V^{\dagger }V=P_{\ker (V)^{\perp }}\). This implies that

$$\begin{aligned} (UV)^{\dagger }x=z=V^{\dagger }Vz {\hbox { and }}Vz=U^{\dagger }UVz=U^{\dagger }x \end{aligned}$$

by (2.8)–(2.10). Thus

$$\begin{aligned} (UV)^{\dagger }x=V^{\dagger }U^{\dagger }x. \end{aligned}$$
(2.11)

Next we prove \(\Vert (UV)^{\dagger }\Vert \le \Vert U^{\dagger }\Vert \Vert V^{\dagger }\Vert \). Arbitrarily fix \(y=y_{1}+y_{2}\in \mathbb {C}^{l}\) with \(y_{1}\in \mathrm{{range}}(UV)\) and \(y_{2}\in \mathrm{{range}}(UV)^{\perp }\). Then \((UV)^{\dagger }y=(UV)^{\dagger }y_{1}\) by the definition of pseudo-inverse. Replacing x in (2.11) by \(y_{1}\) leads to

$$\begin{aligned} \Vert (UV)^{\dagger }y\Vert =&\Vert (UV)^{\dagger }y_{1}\Vert \\ =&\Vert V^{\dagger }U^{\dagger }y_{1}\Vert \\ \le&\Vert U^{\dagger }\Vert \Vert V^{\dagger }\Vert \Vert y_{1}\Vert \\ \le&\Vert U^{\dagger }\Vert \Vert V^{\dagger }\Vert \Vert y\Vert . \end{aligned}$$

This leads to \(\Vert (UV)^{\dagger }\Vert \le \Vert U^{\dagger }\Vert \Vert V^{\dagger }\Vert \) by the arbitrariness of y. \(\square \)

The following lemma shows that \(\sum \limits _{j\in J}c_{j}{\mathcal {G}}_{j}\) is well defined and belongs to \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) whenever \(c\in l^{2}(J)\) and \(\{{\mathcal {G}}_{j}\}_{j\in J}\) is a HS-Bessel sequence.

Lemma 2.2

Let \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) be a HS-Bessel sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with bound B. Then, for all \(c=\{c_{j}\}_{j\in J}\in l^{2}(J)\), \(\Theta =\sum \limits _{j\in J}c_{j}{\mathcal {G}}_{j}\) is well defined, and \(\Theta \in {\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\).

Proof

Arbitrarily fix \(h\in {\mathcal {H}}\). Then

$$\begin{aligned} \left\| \sum _{j\in J}c_{j}{\mathcal {G}}_{j}h\right\| _{2}^{2}\le&\left( \sum _{j\in J}|c_{j}|\Vert {\mathcal {G}}_{j}h\Vert _{2}\right) ^{2}\\ \le&\left( \sum _{j\in J}|c_{j}|^{2}\right) \left( \sum _{j\in J}\Vert {\mathcal {G}}_{j}h\Vert _{2}^{2}\right) \\ \le&B\sum _{j\in J}|c_{j}|^{2}\Vert h\Vert ^{2} \end{aligned}$$

for \(c\in l_{0}(J)\), where B is the HS-Bessel bound of \({\mathcal {G}}\). This leads to the lemma due to \(l_{0}(J)\) being dense in \(l^{2}(J)\). \(\square \)

The following lemma shows that a bounded linear operator on \(l^{2}(J)\) induces a bounded linear operator on \(\oplus {\mathcal {C}}_{2}\).

Lemma 2.3

Suppose \(U=(u_{k,\,j})_{k,\,j\in J}\in {\mathcal {B}}(l^{2}(J))\). Define \(\Lambda :\oplus {\mathcal {C}}_{2}\rightarrow \oplus {\mathcal {C}}_{2}\) by

$$\begin{aligned} \Lambda {\mathcal {A}}=\left\{ \sum _{j\in J}\overline{u_{k,\,j}}{\mathcal {A}}_{j}\right\} _{k\in J}{\hbox { }}\left( {\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\right) . \end{aligned}$$

Then \(\Lambda \) is bounded on \(\oplus {\mathcal {C}}_{2}.\)

Proof

First we claim that \(\sum \limits _{j\in J}c_{j}{\mathcal {A}}_{j}\) is well defined if \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\) and \(c=\{c_{j}\}_{j\in J}\in l^{2}\). Indeed, for an arbitrary finite subset \({\widetilde{J}}\) of J,

$$\begin{aligned} \left\| \sum \limits _{j\in {\widetilde{J}}}c_{j}{\mathcal {A}}_{j}\right\| _{2} \le \left( \sum \limits _{j\in {\widetilde{J}}}|c_{j}|^{2}\right) ^{\frac{1}{2}}\left( \sum \limits _{j\in {\widetilde{J}}}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}\right) ^{\frac{1}{2}}\le \Vert {\mathcal {A}}\Vert _{2}\left( \sum \limits _{j\in {\widetilde{J}}}|c_{j}|^{2}\right) ^{\frac{1}{2}}. \end{aligned}$$

This implies that \(\sum \limits _{j\in J}c_{j}{\mathcal {A}}_{j}\) converges unconditionally. Since \(U\in {\mathcal {B}}(l^{2}(J))\), we have \(U^{*}\in {\mathcal {B}}(l^{2}(J))\). It follows that each column \(\{\overline{u_{k,\,j}}\}_{j\in J}\) of \(U^{*}\) belongs to \(l^{2}\). Thus, for each \(k\in J\), \(\sum \limits _{j\in J}\overline{u_{k,\,j}}{\mathcal {A}}_{j}\) is well defined. Next we prove that \(\Lambda \) is a bounded operator on \(\oplus {\mathcal {C}}_{2}.\)

Let \(\{\xi _{m}\}_{m\in M}\) be an orthonormal basis for \({\mathcal {C}}_{2}\). Arbitrarily fix \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0} {\mathcal {C}}_{2}\), then there exists a sequence \(\{c_{m}^{(j)}\}_{j\in J,\,m\in M}\in l^{2}\) such that \({\mathcal {A}}_{j}=\sum \limits _{m\in M}c_{m}^{(j)}\xi _{m}\) for \(j\in J\), where \(\{c_{m}^{(j)}\}_{j\in J}\in l_{0}\) for each \(m\in M\). It follows that

$$\begin{aligned} \Vert {\mathcal {A}}\Vert _{2}^{2}=\sum _{m\in M}\sum _{j\in J}\left| c_{m}^{(j)}\right| ^{2}, \end{aligned}$$

and

$$\begin{aligned} \Vert \Lambda {\mathcal {A}}\Vert _{2}^{2} =&\sum _{k\in J}\left\| \sum _{j\in J}\overline{u_{k,\,j}}\sum \limits _{m\in M}c_{m}^{(j)}\xi _{m}\right\| ^{2}_{2}\\ =&\sum _{k\in J}\sum \limits _{m\in M}\left| \sum _{j\in J}\overline{u_{k,\,j}}c_{m}^{(j)}\right| ^{2}\\ =&\sum \limits _{m\in M}\left\| {\overline{U}}\{c_{m}^{(j)}\}_{j\in J}\right\| ^{2}. \end{aligned}$$

Thus

$$\begin{aligned} \Vert \Lambda {\mathcal {A}}\Vert _{2}^{2}\le \Vert U\Vert ^{2}\sum _{m\in M}\sum _{j\in J}\left| c_{m}^{(j)}\right| ^{2}= \Vert U\Vert ^{2}\Vert {\mathcal {A}}\Vert _{2}^{2}. \end{aligned}$$

Therefore, \(\Lambda \) is bounded on \(\oplus {\mathcal {C}}_{2}\) due to \(\oplus _{0}{\mathcal {C}}_{2}\) being dense in \(\oplus {\mathcal {C}}_{2}\). \(\square \)

The following lemma shows that every \(l^{2}(J)\)-decomposable sequence in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) is a HS-Bessel sequence.

Lemma 2.4

An arbitrary \(l^{2}(J)\)-decomposable sequence in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) is a HS-Bessel sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\).

Proof

Suppose \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) is an \(l^{2}(J)\)-decomposable sequence with respect to a HS-orthonormal basis \({\mathcal {E}}=\{{\mathcal {E}}_{j}\}_{j\in J}\), and

$$\begin{aligned} {\mathcal {G}}_{j}=\sum \limits _{k\in J}u_{k,\,j}{\mathcal {E}}_{k}{\hbox { }}(j\in J), \end{aligned}$$

where \(U=(u_{k,\,j})_{k,\,j\in J}\in {\mathcal {B}}(l^{2}(J))\). Then

$$\begin{aligned} {\mathcal {G}}_{j}^{*}=\sum \limits _{k\in J}\overline{u_{k,\,j}}{\mathcal {E}}_{k}^{*}{\hbox { }}(j\in J). \end{aligned}$$
(2.12)

Next we prove \({\mathcal {G}}\) is a HS-Bessel sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\). Arbitrarily fix \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0} {\mathcal {C}}_{2}\). Then

$$\begin{aligned} \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}=\sum _{j\in J}\sum _{k\in J}\overline{u_{k,\,j}}{\mathcal {E}}_{k}^{*}{\mathcal {A}}_{j}=\sum _{k\in J}{\mathcal {E}}_{k}^{*}\left( \sum _{j\in J}\overline{u_{k,\,j}}{\mathcal {A}}_{j}\right) \end{aligned}$$
(2.13)

by (2.12). Let \(\{\xi _{m}\}_{m\in M}\) be an orthonormal basis for \({\mathcal {C}}_{2}\). Then there exists a sequence \(\{c_{m}^{(j)}\}_{j\in J,\,m\in M}\in l^{2}\) such that \({\mathcal {A}}_{j}=\sum \limits _{m\in M}c_{m}^{(j)}\xi _{m}\) for \(j\in J\), where \(\{c_{m}^{(j)}\}_{j\in J}\in l_{0}\) for each \(m\in M\). It follows that

$$\begin{aligned} \sum _{m\in M}\left\| {\overline{U}}\{c_{m}^{(j)}\}_{j\in J}\right\| ^{2}\le \Vert U\Vert ^{2}\sum _{m\in M}\sum _{j\in J}|c_{m}^{(j)}|^{2}=\Vert U\Vert ^{2}\Vert {\mathcal {A}}\Vert _{2}^{2}. \end{aligned}$$
(2.14)

Observe that

$$\begin{aligned} \sum _{m\in M}\left\| {\overline{U}}\{c_{m}^{(j)}\}_{j\in J}\right\| ^{2}= & {} \sum _{k\in J}\sum _{m\in M}\left| \sum _{j\in J}\overline{u_{k,\,j}}c_{m}^{(j)}\right| ^{2}\nonumber \\= & {} \sum _{k\in J}\left\| \sum _{m\in M}\left( \sum _{j\in J}\overline{u_{k,\,j}}c_{m}^{(j)}\right) \xi _{m}\right\| _{2}^{2}\nonumber \\= & {} \left\| \sum _{k\in J}{\mathcal {E}}_{k}^{*}\sum \limits _{m\in M}\left( \sum _{j\in J}\overline{u_{k,\,j}}c_{m}^{(j)}\right) \xi _{m}\right\| ^{2}\nonumber \\= & {} \left\| \sum _{k\in J}{\mathcal {E}}_{k}^{*}\left( \sum _{j\in J}\overline{u_{k,\,j}}{\mathcal {A}}_{j}\right) \right\| ^{2} \end{aligned}$$
(2.15)

by [16, Lemma 2.1] due to \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0}{\mathcal {C}}_{2}\). Collecting (2.13)-(2.15), we obtain that

$$\begin{aligned} \left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}\right\| ^{2}\le \Vert U\Vert ^{2}\Vert {\mathcal {A}}\Vert _{2}^{2}{\hbox { }}\left( {\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0}{\mathcal {C}}_{2}\right) . \end{aligned}$$

This implies that \({\mathcal {G}}\) is a HS-Bessel sequence due to \(\oplus _{0}{\mathcal {C}}_{2}\) being dense in \(\oplus {\mathcal {C}}_{2}\). \(\square \)

The following lemma focuses on HS-frame property of \(l^{2}(J)\)-decomposable HS-operator sequences.

Lemma 2.5

Given a HS-orthonormal basis \({\mathcal {E}}=\{{\mathcal {E}}_{j}\}_{j\in J}\) for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\), let \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) be an \(l^{2}(J)\)-decomposable sequence in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) with respect to \({\mathcal {E}}\) associated with \(U=(u_{k,\,j})_{k,\,j\in J}\in {\mathcal {B}}\left( l^{2}(J)\right) \), i.e.,

$$\begin{aligned} {\mathcal {G}}_{j}=\sum _{k\in J}u_{k,\,j}{\mathcal {E}}_{k}{\hbox { }}(j\in J). \end{aligned}$$

Then \({\mathcal {G}}\) is a HS-frame (HS-Riesz basis, HS-frame sequence, HS-Riesz sequence) for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) if U is bounded and surjective (bijective, of closed range, injective and of closed range) on \(l^{2} (J)\).

Proof

By Lemma 2.4, \({\mathcal {G}}\) is a HS-Bessel sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\). By a simple computation,

$$\begin{aligned} T_{{\mathcal {G}}}{\mathcal {A}}=T_{{\mathcal {E}}}\Lambda {\mathcal {A}}{\hbox { }}\left( {\mathcal {A}}\in \oplus _{0} {\mathcal {C}}_{2}\right) , \end{aligned}$$

where \(\Lambda \) is as in Lemma 2.3. Observe that \(\Lambda \) is bounded by Lemma 2.3, and \(T_{{\mathcal {G}}}\), \(T_{{\mathcal {E}}}\) are also bounded. It follows that

$$\begin{aligned} T_{{\mathcal {G}}}{\mathcal {A}}=T_{{\mathcal {E}}}\Lambda {\mathcal {A}}{\hbox { }}\left( {\mathcal {A}}\in \oplus {\mathcal {C}}_{2}\right) \end{aligned}$$
(2.16)

due to \(\oplus _{0} {\mathcal {C}}_{2}\) being dense in \(\oplus {\mathcal {C}}_{2}\). By [32, Proposition 2.13 and Theorem 2.15] and [16, Lemmas 2.5 and 2.6], we need to prove that \(T_{{\mathcal {G}}}\) is surjective (bijective, of closed range, injective and of closed range) when U is surjective (bijective, of closed range, injective and of closed range). Since \(T_{{\mathcal {E}}}\) is unitary, by (2.16), \(T_{{\mathcal {G}}}\) is surjective (bijective, of closed range, injective and of closed range) if and only if \(\Lambda \) is surjective (bijective, of closed range, injective and of closed range). So it is enough to prove that \(\Lambda \) is surjective (injective, of closed range) if U is surjective (injective, of closed range). Let \(\{\xi _{m}\}_{m\in M}\) be an orthonormal basis for \({\mathcal {C}}_{2}\). Then

$$\begin{aligned} {\mathcal {D}}_{j}=\sum _{m\in M}\langle {\mathcal {D}}_{j},\,\xi _{m}\rangle _{\tau }\xi _{m}{\hbox { }}(j\in J), \end{aligned}$$
(2.17)

and

$$\begin{aligned} \Lambda {\mathcal {D}}=\left\{ \sum _{m\in M}\left( \sum _{j\in J}\overline{u_{k,\,j}}\langle {\mathcal {D}}_{j},\xi _{m}\rangle _{\tau }\right) \xi _{m}\right\} _{k\in J}{\hbox { }}\left( {\mathcal {D}}=\{{\mathcal {D}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\right) . \end{aligned}$$
(2.18)

Case 1. U is surjective.

Arbitrarily fix \({\mathcal {B}}=\{{\mathcal {B}}_{k}\}_{k\in J}\in \oplus {\mathcal {C}}_{2}\). Then

$$\begin{aligned} \Vert {\mathcal {B}}\Vert _{2}^{2}=\sum _{k\in J}\Vert {\mathcal {B}}_{k}\Vert _{2}^{2}=\sum _{k\in J}\sum _{m\in M}\left| \langle {\mathcal {B}}_{k},\xi _{m}\rangle _{\tau }\right| ^{2}<\infty . \end{aligned}$$
(2.19)

Define \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\) by

$$\begin{aligned} {\mathcal {A}}_{j}=\sum _{m\in M}c_{m}^{(j)}\xi _{m}{\hbox { }}(j\in J) \end{aligned}$$

where

$$\begin{aligned} \{c_{m}^{(j)}\}_{j\in J}={\overline{U}}^{\dagger }\left( \left\{ \langle {\mathcal {B}}_{j},\,\xi _{m}\rangle _{\tau }\right\} _{j\in J}\right) {\hbox { }}(m\in M). \end{aligned}$$

Then

$$\begin{aligned} \sum _{m\in M}\sum _{j\in J}\left| c_{m}^{(j)}\right| ^{2}\le \Vert {\overline{U}}^{\dagger }\Vert ^{2}\sum _{m\in M}\sum _{j\in J}\left| \langle {\mathcal {B}}_{j},\,\xi _{m}\rangle _{\tau }\right| ^{2}<\infty \end{aligned}$$

by (2.19). So \({\mathcal {A}}\) is well defined, and

$$\begin{aligned} \Vert {\mathcal {A}}\Vert _{2}^{2}\le \Vert {\overline{U}}^{\dagger }\Vert ^{2}\Vert {\mathcal {B}}\Vert _{2}^{2}. \end{aligned}$$

By (2.17), (2.18) and a simple computation, we have

$$\begin{aligned} \Lambda {\mathcal {A}}=\left\{ \sum _{m\in M}d_{m}^{(k)}\xi _{m}\right\} _{k\in J} \end{aligned}$$

with

$$\begin{aligned} \left\{ d_{m}^{(k)}\right\} _{k\in J}={\overline{U}}\,{\overline{U}}^{\dagger }\left\{ \langle {\mathcal {B}}_{k},\xi _{m}\rangle _{\tau }\right\} _{k\in J}. \end{aligned}$$

Observing \({\overline{U}}\,{\overline{U}}^{\dagger }=I\) leads to \(\Lambda {\mathcal {A}}={\mathcal {B}}\). Thus \(\Lambda \) is surjective.

Case 2. U is injective.

Suppose \(\Lambda {\mathcal {A}}=0\) for some \({\mathcal {A}}\in \oplus {\mathcal {C}}_{2}\). Then

$$\begin{aligned} \Vert {\mathcal {A}}\Vert _{2}^{2}=\sum _{j\in J}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}=\sum _{j\in J}\sum _{m\in M}\left| \langle {\mathcal {A}}_{j},\xi _{m}\rangle _{\tau }\right| ^{2}<\infty \end{aligned}$$

and

$$\begin{aligned} {\overline{U}}\left( \{\langle {\mathcal {A}}_{j},\xi _{m}\rangle _{\tau }\}_{j\in J}\right) =0{\hbox { }}(m\in M) \end{aligned}$$

by (2.17) and (2.18). This leads to

$$\begin{aligned} \langle {\mathcal {A}}_{j},\xi _{m}\rangle _{\tau }=0{\hbox { }}(j\in J{\hbox { and }}m\in M), \end{aligned}$$

due to \({\overline{U}}\) being injective. Equivalently, \({\mathcal {A}}=0\). Thus \(\Lambda \) is injective.

Case 3. U is of closed range.

Suppose there exists a sequence \(\{{\mathcal {A}}^{(n)}\}_{n\in \mathbb {N}}\subset \oplus {\mathcal {C}}_{2}\) with \({\mathcal {A}}^{(n)}=\{{\mathcal {A}}_{j}^{(n)}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\) for \(n\in \mathbb {N}\) such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Lambda {\mathcal {A}}^{(n)}={\mathcal {B}}{ \text{ for } \text{ some } }{\mathcal {B}}=\{{\mathcal {B}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}. \end{aligned}$$
(2.20)

Then

$$\begin{aligned} \left\| \Lambda {\mathcal {A}}^{(n)}-{\mathcal {B}}\right\| ^{2} =&\left\| \left\{ \sum \limits _{m\in M}\sum _{j\in J}\overline{u_{k,\,j}}\langle {\mathcal {A}}_{j}^{(n)},\,\xi _{m}\rangle _{\tau }\xi _{m}-\sum \limits _{m\in M}\langle {\mathcal {B}}_{k},\,\xi _{m}\rangle _{\tau }\xi _{m}\right\} _{k\in J}\right\| ^{2}\nonumber \\ =&\sum _{m\in M}\left\| {\overline{U}}\{\langle {\mathcal {A}}_{j}^{(n)},\,\xi _{m}\rangle _{\tau }\}_{j\in J}-\{\langle {\mathcal {B}}_{k},\,\xi _{m}\rangle _{\tau }\}_{k\in J}\right\| ^{2} \end{aligned}$$
(2.21)

by (2.17), (2.18) due to \(\{\xi _{m}\}_{m\in M}\) being an orthonormal basis for \({\mathcal {C}}_{2}\). Collecting (2.20) and (2.21) leads to

$$\begin{aligned} \lim _{n\rightarrow \infty }{\overline{U}}\{\langle {\mathcal {A}}_{j}^{(n)},\,\xi _{m}\rangle _{\tau }\}_{j\in J}=\{\langle {\mathcal {B}}_{k},\,\xi _{m}\rangle _{\tau }\}_{k\in J}{\hbox { }}(m\in M). \end{aligned}$$

Since U is of closed range, so is \({\overline{U}}\). Then, for each \(m\in M\), there exists a \(\{c_{m}^{(j)}\}_{j\in J}\in \ker ({\overline{U}})^{\perp }\) such that

$$\begin{aligned} {\overline{U}}\{c_{m}^{(j)}\}_{j\in J}=\{\langle {\mathcal {B}}_{k},\,\xi _{m}\rangle _{\tau }\}_{k\in J}, \end{aligned}$$

equivalently,

$$\begin{aligned} \{c_{m}^{(j)}\}_{j\in J}={\overline{U}}^{\dagger }\{\langle {\mathcal {B}}_{k},\,\xi _{m}\rangle _{\tau }\}_{k\in J}{\hbox { }}(m\in M). \end{aligned}$$

Define \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}\) by

$$\begin{aligned} {\mathcal {A}}_{j}=\sum \limits _{m\in M}c_{m}^{(j)}\xi _{m}{\hbox { }}(j\in J). \end{aligned}$$

Observe that \({\overline{U}}\,{\overline{U}}^{\dagger }=P_{\mathrm{{range}}({\overline{U}})}\). Similarly to Case 1, we have \({\mathcal {A}}\in \oplus {\mathcal {C}}_{2}\) and \(\Lambda {\mathcal {A}}={\mathcal {B}}\). Thus \(\Lambda \) has closed range. The proof is completed. \(\square \)

The following theorem shows that a bounded invertible operator on \(l^{2}(J)\) transforms a HS-frame into another HS-frame.

Theorem 2.1

Let \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) be a HS-frame for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\). Suppose \(\Gamma =(\gamma _{k,\,j})_{k,\,j\in J}\) is a bounded operator on \(l^{2}(J)\). Define \({\mathcal {F}}_{j}=\sum \limits _{k\in J}\gamma _{k,\,j}{\mathcal {G}}_{k}\) for \(k\in J\). Then \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) is a HS-frame for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) when \(\Gamma \) is invertible on \(l^{2}(J)\).

Proof

By Lemma 2.2, \({\mathcal {F}}_{j}\) is well defined and \({\mathcal {F}}_{j}\in {\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) for \(j\in J\). Let \(\{\xi _{m}\}_{m\in M}\) be an orthonormal basis for \({\mathcal {C}}_{2}\). Then

$$\begin{aligned} \sum _{j\in J}\left\| {\mathcal {F}}_{j}h\right\| _{2}^{2} =&\sum _{j\in J}\sum _{m\in M}\left| \left\langle \sum _{k\in J}\gamma _{k,\,j} {\mathcal {G}}_{k}h,\,\xi _{m}\right\rangle _{\tau }\right| ^{2}\\ =&\sum _{m\in M}\sum _{j\in J}\left| \sum _{k\in J}\gamma _{k,\,j}\left\langle {\mathcal {G}}_{k}h,\,\xi _{m}\right\rangle _{\tau }\right| ^{2} \end{aligned}$$

for \(h\in {\mathcal {H}}\). Also observe that \(\sum \limits _{k\in J}\left| \langle {\mathcal {G}}_{k}h,\,\xi _{m}\rangle _{\tau }\right| ^{2}\le \sum \limits _{k\in J}\Vert {\mathcal {G}}_{k}h\Vert _{2}^{2}<\infty .\) It follows that

$$\begin{aligned} \sum _{j\in J}\left\| {\mathcal {F}}_{j}h\right\| _{2}^{2} =\sum _{m\in M}\left\| \Gamma \big (\left\{ \langle {\mathcal {G}}_{k}h,\,\xi _{m}\right\rangle _{\tau }\}_{k\in J}\big )\right\| ^{2}{\hbox { }}(h\in {\mathcal {H}}). \end{aligned}$$
(2.22)

When \(\Gamma \) is invertible, there exist constants \(0<c_{1}\le c_{2}<\infty \) such that

$$\begin{aligned} c_{1}\sum \limits _{k\in J}\left| \langle {\mathcal {G}}_{k}h,\,\xi _{m}\rangle _{\tau }\right| ^{2}\le \left\| \Gamma (\left\{ \langle {\mathcal {G}}_{k}h,\,\xi _{m}\right\rangle _{\tau }\}_{k\in J})\right\| ^{2}\le c_{2}\sum \limits _{k\in J}\left| \langle {\mathcal {G}}_{k}h,\,\xi _{m}\rangle _{\tau }\right| ^{2}{\hbox { }}(h\in {\mathcal {H}}). \end{aligned}$$

This together with (2.22) leads to

$$\begin{aligned} c_{1}\sum _{k\in J}\left\| {\mathcal {G}}_{k}h\right\| _{2}^{2}\le \sum _{j\in J}\left\| {\mathcal {F}}_{j}h\right\| _{2}^{2}\le c_{2}\sum _{k\in J}\left\| {\mathcal {G}}_{k}h\right\| _{2}^{2}{\hbox { }}(h\in {\mathcal {H}}). \end{aligned}$$

Thus \({\mathcal {F}}\) is a HS-frame due to \({\mathcal {G}}\) being a HS-frame. The proof is completed. \(\square \)

Remark 2.1

In Theorem 2.1, if \({\mathcal {G}}\) is a HS-frame sequence, then replacing \({\mathcal {H}}\) by \(\overline{\mathrm{span}}\{{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}\}_{j\in J}\) leads to \({\mathcal {F}}\) being a HS-frame sequence with

$$\begin{aligned} \overline{\mathrm{span}}\{{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}\}_{j\in J}=\overline{\mathrm{span}}\{{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}\}_{j\in J}. \end{aligned}$$

The following theorem characterizes the HS-frame property of the \(l^{2}(J)\)-operator portraits of HS-frames.

Theorem 2.2

Let \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) be an \(l^{2}(J)\)-decomposable sequence in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) with respect to a HS-orthonormal basis \({\mathcal {E}}=\{{\mathcal {E}}_{j}\}_{j\in J}\) associated with \(U=(u_{j,\,k})_{j,k\in J}\in {\mathcal {B}}(l^{2}(J))\), i.e.,

$$\begin{aligned} {\mathcal {G}}_{j}=\sum \limits _{k\in J}u_{k,\,j}{\mathcal {E}}_{k}{\hbox { }}(j\in J). \end{aligned}$$

Suppose \({\mathcal {G}}\) is a HS-frame for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\), and \(V=(v_{j,k})_{j,k\in J}\in {\mathcal {B}}(l^{2}(J))\). Let \({\mathcal {F}}_{j}=\sum \limits _{k\in J}v_{k,\,j}{\mathcal {G}}_{k}\) for \(j\in J\). Then \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) is a HS-frame (HS-Riesz basis, HS-frame sequence, HS-Riesz sequence) for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) if UV is bounded and surjective (bijective, of closed range, injective and of closed range) on \(l^{2} (J)\).

Proof

By a standard argument, we have \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) is an \(l^{2}(J)\)-decomposable sequence in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\) with respect to \({\mathcal {E}}\) associated with \(UV\in {\mathcal {B}}(l^{2}(J))\). Applying Lemma 2.5 to UV gives the theorem. \(\square \)

The following is an example of Theorem 2.2.

Example 2.2

Let \({\mathcal {E}}=\{{\mathcal {E}}_{j}\}_{j\in \mathbb {N}}\) be a HS-orthonormal basis for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\). Define \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in \mathbb {N}}\) by

$$\begin{aligned} ({\mathcal {G}}_{n_{k-1}+1},\,{\mathcal {G}}_{n_{k-1}+2},\,\cdots ,{\mathcal {G}}_{n_{k}})=({\mathcal {E}}_{l_{k-1}+1},\,{\mathcal {E}}_{l_{k-1}+2},\,\cdots ,{\mathcal {E}}_{l_{k}})U_{k} {\hbox { }}(k\in \mathbb {N}), \end{aligned}$$
$$\begin{aligned} \sup \limits _{k\in \mathbb {N}}\Vert U_{k}\Vert<\infty {\hbox { and }}\sup \limits _{k\in \mathbb {N}}\Vert U^{\dagger }_{k}\Vert <\infty , \end{aligned}$$

where \(U_{k}\) are \((l_{k}-l_{k-1})\times (n_{k}-n_{k-1})\) matrices such that \(rank(U_{k})=l_{k}-l_{k-1}\) for \(k\in \mathbb {N}\) (write \(l_{0}=n_{0}=0\)). Then

$$\begin{aligned} ({\mathcal {G}}_{1},\,{\mathcal {G}}_{2},\,\cdots )=({\mathcal {E}}_{1},\,{\mathcal {E}}_{2},\,\cdots )U{\hbox { with }}U=diag(U_{1}, U_{2}, \cdots ). \end{aligned}$$

Obviously, \(\Vert U\Vert \le \sup \limits _{k\in \mathbb {N}}\Vert U_{k}\Vert \). Arbitrarily fix \(Y=(Y_{k})_{k\in \mathbb {N}}\in l^{2}\) with \(Y_{k}\in \mathbb {C}^{l_{k}-l_{k-1}}\). Since \(rank(U_{k})=l_{k}-l_{k-1}\), we have \(U_{k}U_{k}^{\dagger }Y_{k}=Y_{k}\) for \(k\in \mathbb {N}\). Take \(X=\left( U_{k}^{\dagger }Y_{k}\right) _{k\in \mathbb {N}}\). Then

$$\begin{aligned} \Vert X\Vert \le \left( \sup _{k\in \mathbb {N}}\Vert U_{k}^{\dagger }\Vert \right) \Vert Y\Vert <\infty , \end{aligned}$$

and

$$\begin{aligned} UX=Y. \end{aligned}$$

Thus U is bounded and surjective on \(l^{2}(\mathbb {N})\). It follows that \({\mathcal {G}}\) is a HS-frame for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) by Lemma 2.5.

Define matrix \(V=diag(V_{1}, V_{2}, \cdots )\) satisfying

$$\begin{aligned} \sup \limits _{k\in \mathbb {N}}\Vert V_{k}\Vert<\infty {\hbox { and }}\sup \limits _{k\in \mathbb {N}}\Vert V_{k}^{\dagger }\Vert <\infty , \end{aligned}$$

where \(V_{k}\) are \((n_{k}-n_{k-1})\times (m_{k}-m_{k-1})\) matrices such that \(\mathrm{{range}}(V_{k})\subset \mathrm{{range}}(U_{k}^{*})\) for \(k\in \mathbb {N}\) with \(l_{0}=m_{0}=0\). Let \({\mathcal {F}}_{j}=\sum \limits _{k\in J}v_{j,k}{\mathcal {G}}_{k}\) for \(j\in \mathbb {N}\). Then we have the following claims.

Claim 1. UV is bounded on \(l^{2}(\mathbb {N})\), and UV is surjective (injective) if and only if each \(U_{k}V_{k}\) with \(k\in \mathbb {N}\) is surjective (injective).

Obviously, \(\Vert UV\Vert \le \left( \sup \limits _{k\in \mathbb {N}}\Vert U_{k}\Vert \right) \left( \sup \limits _{k\in \mathbb {N}}\Vert V_{k}\Vert \right) <\infty \), UV is injective if and only if each \(U_{k}V_{k}\) with \(k\in \mathbb {N}\) is injective. Suppose each \(U_{k}V_{k}\) with \(k\in \mathbb {N}\) is surjective. Arbitrarily fix \(Y=(Y_{k})_{k\in \mathbb {N}}\) in \(l^{2}\) with \(Y_{k}\in \mathbb {C}^{l_{k}-l_{k-1}}\). Since \(U_{k}V_{k}\) is surjective and \(\mathrm{{range}}(V_{k})\subset \mathrm{{range}}(U_{k}^{*})\), we have that

$$\begin{aligned} U_{k}V_{k}(U_{k}V_{k})^{\dagger }Y_{k}=Y_{k}, \end{aligned}$$

and

$$\begin{aligned} \Vert (U_{k}V_{k})^{\dagger }Y_{k}\Vert =\Vert V_{k}^{\dagger }U_{k}^{\dagger }Y_{k}\Vert \le \Vert U_{k}^{\dagger }\Vert \Vert V_{k}^{\dagger }\Vert \Vert Y_{k}\Vert \end{aligned}$$
(2.23)

for \(k\in \mathbb {N}\) by Lemma 2.1. Take \(X=(X_{k})_{k\in \mathbb {N}}\) with \(X_{k}=(U_{k}V_{k})^{\dagger }Y_{k}\). Then

$$\begin{aligned} \Vert X\Vert ^{2}\le \left( \sup _{k\in \mathbb {N}}\Vert U_{k}^{\dagger }\Vert \right) ^{2}\left( \sup _{k\in \mathbb {N}}\Vert V_{k}^{\dagger }\Vert \right) ^{2}\Vert Y\Vert ^{2}<\infty \end{aligned}$$

by (2.23), and

$$\begin{aligned} UVX=Y. \end{aligned}$$

Thus UV is surjective. Obviously, UV being surjective implies each \(U_{k}V_{k}\) with \(k\in \mathbb {N}\) being surjective. Claim 1 therefore follows.

Claim 2. UV is of closed range. Suppose \(\{X^{(n)}\}_{n\in \mathbb {N}}\) is a sequence in \(l^{2}\) such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| UVX^{(n)}-Y\right\| =0 \end{aligned}$$

for some \(Y\in l^{2}\), where \(X^{(n)}=\left( X_{k}^{(n)}\right) _{k\in \mathbb {N}}\), \(Y=(Y_{k})_{k\in \mathbb {N}}\) with \(X_{k}^{(n)}\in \mathbb {C}^{m_{k}-m_{k-1}},\) \(Y_{k}\in \mathbb {C}^{l_{k}-l_{k-1}}\). Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| U_{k}V_{k}X_{k}^{(n)}-Y_{k}\right\| =0{\hbox { }}(k\in \mathbb {N}). \end{aligned}$$

Since each \(U_{k}V_{k}\) is a \((l_{k}-l_{k-1})\times (m_{k}-m_{k-1})\) matrix, it is of closed range. Take \(X_{k}=(U_{k}V_{k})^{\dagger }Y_{k}\) for \(k\in \mathbb {N}\). Then

$$\begin{aligned} U_{k}V_{k}X_{k}=Y_{k}{\hbox { }}(k\in \mathbb {N}). \end{aligned}$$

Let \(X=(X_{k})_{k\in \mathbb {N}}\). Then \(X\in l^{2}\) similarly to Claim 1. Obviously, \(UVX=Y\). Thus UV is of closed range.

By Theorem 2.2 and Claims 1 and 2, we have

(i) Suppose \(m_{k}= l_{k}\) and \(rank(U_{k}V_{k})=l_{k}-l_{k-1}\) for \(k\in \mathbb {N}\). Then \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in \mathbb {N}}\) is a HS-Riesz basis for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\).

(ii) Suppose \(rank(U_{k}V_{k})=l_{k}-l_{k-1}\) for \(k\in \mathbb {N}\), and there exists at least one \(k_{0}\in \mathbb {N}\) such that \(m_{k_{0}}-m_{k_{0}-1}>l_{k_{0}}-l_{k_{0}-1}\). Then \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in \mathbb {N}}\) is a HS-frame but not a HS-Riesz basis for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\).

(iii) Suppose there exists at least one \(k_{0}\in \mathbb {N}\) such that \(rank(U_{k_{0}}V_{k_{0}})<l_{k_{0}}-l_{k_{0}-1}\). Then \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in \mathbb {N}}\) is a HS-frame sequence but not a HS-frame for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\).

(iv) Suppose \(rank(U_{k}V_{k})=m_{k}-m_{k-1}\) for each \(k\in \mathbb {N}\), and there exists at least one \(k_{0}\in \mathbb {N}\) such that \(m_{k_{0}}-m_{k_{0}-1}<l_{k_{0}}-l_{k_{0}-1}\). Then \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in \mathbb {N}}\) is a HS-Riesz sequence but not a HS-Riesz basis for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\).

3 Perturbation of HS-Frame Sequences

This section focuses on stable perturbations of HS-frame sequences (HS-Riesz sequences). For this purpose, we quote following lemmas from [11, 14, 26].

Lemma 3.1

[26] Let \({\mathcal {H}}_{1}\) and \({\mathcal {H}}_{2}\) be two Hilbert spaces, and \(U\in {\mathcal {B}}({\mathcal {H}}_{1},{\mathcal {H}}_{2})\). Then

(i) \(\gamma (U)=\inf \Big \{\Vert Uh\Vert :h\in (\ker U)^{\perp },\Vert h\Vert =1 \Big \}>0\) if and only if \(\mathrm{{range}}(U)\) is closed.

(ii) If \(\mathrm{{range}}(U)\) is closed, then \(\Vert U^{\dagger }\Vert =\gamma (U)^{-1}\).

Lemma 3.2

[11, Theorem 2.2] Given Hilbert spaces \({\mathcal {H}}_{1}\) and \( {\mathcal {H}}_{2}\), and let \(T,\,U\in {\mathcal {B}}({\mathcal {H}}_{1},{\mathcal {H}}_{2})\). Suppose that \(\delta =\delta \left( \ker (T),\ker (U)\right) <1\), and there exist numbers \(0\le \lambda _{1}<1\), \(\lambda _{2}>-1\) and \(\mu \ge 0\) such that

$$\begin{aligned} \Vert Th-Uh\Vert \le \lambda _{1}\Vert Th\Vert +\lambda _{2}\Vert Uh\Vert +\mu \Vert h\Vert {\hbox { }}(h\in {\mathcal {H}}_{1}). \end{aligned}$$

Then

(i) \(\gamma (U)\ge \frac{(1-\lambda _{1})\gamma (T)\sqrt{1-\delta ^{2}}-\mu }{1+\lambda _{2}}\).

(ii) If \(\mathrm{{range}}(T)\) is closed and \(\lambda _{1}+\frac{\mu }{\gamma (T)\sqrt{1-\delta ^{2}}}<1\), then \(\mathrm{{range}}(U)\) is closed and

$$\begin{aligned} \Vert U^{\dagger }\Vert \le \frac{(1+\lambda _{2})\Vert T^{\dagger }\Vert }{(1-\lambda _{1})\sqrt{1-\delta ^{2}}-\mu \Vert T^{\dagger }\Vert }. \end{aligned}$$

Lemma 3.3

[14, Lemma 2.1] Let \({\mathcal {H}}\) be a Hilbert space. Then \(\delta (V,W)=\Vert P_{V}P_{W^{\perp }}\Vert \) for all closed subspaces V and W of \({\mathcal {H}}\).

The following theorem gives a perturbation theorem of HS-frame sequences, where the gap between the kernel spaces of synthesis operators is involved.

Theorem 3.1

Let \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) be a HS-frame sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with bounds A and B. Suppose that \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) is a sequence in \({\mathcal {B}}({\mathcal {H}},{\mathcal {C}}_{2})\), and that there exist constants \(\lambda _{2}\in [0,1)\) and \(\lambda _{1}\), \(\mu \ge 0\) such that

$$\begin{aligned} \left\| \sum _{j\in J}({\mathcal {G}}_{j}^{*}-{\mathcal {F}}_{j}^{*}){\mathcal {A}}_{j}\right\| \le \lambda _{1}\left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}\right\| +\lambda _{2}\left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}\right\| +\mu \left( \sum _{j\in J}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}\right) ^{\frac{1}{2}} \end{aligned}$$
(3.1)

for \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0}{\mathcal {C}}_{2}\). Then \({\mathcal {F}}\) is a HS-Bessel sequence with bound \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\).

If furthermore \(\delta =\delta \left( \ker (T_{{\mathcal {G}}}),\,\ker (T_{{\mathcal {F}}})\right) <1\) and \(\lambda _{1}+\frac{\mu }{\sqrt{A(1-\delta ^{2})}}<1\), then \({\mathcal {F}}\) is a HS-frame sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with lower bound \(\left( \frac{(1-\lambda _{1})\sqrt{A(1-\delta ^{2})}-\mu }{1+\lambda _{2}}\right) ^{2}\).

Proof

By the same procedure as in [32, Theorem 4.3], \({\mathcal {F}}\) is a HS-Bessel sequence with bound \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\). It follows that

$$\begin{aligned} \left\| T_{{\mathcal {G}}}{\mathcal {A}}-T_{{\mathcal {F}}}{\mathcal {A}}\right\| \le \lambda _{1}\left\| T_{{\mathcal {G}}}{\mathcal {A}}\right\| +\lambda _{2}\left\| T_{{\mathcal {F}}}{\mathcal {A}}\right\| +\mu \Vert {\mathcal {A}}\Vert {\hbox { }}\left( {\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\right) \end{aligned}$$
(3.2)

by (3.1) due to \(\oplus _{0}{\mathcal {C}}_{2}\) being dense in \(\oplus {\mathcal {C}}_{2}\). Next we prove that \({\mathcal {F}}\) satisfies the lower bound condition if \(\delta <1\) and \(\lambda _{1}+\frac{\mu }{\sqrt{A(1-\delta ^{2})}}<1\). Since \({\mathcal {G}}\) is a HS-frame sequence, \(\mathrm{{range}}(T_{\mathcal {G}})\) is closed by [16, Lemma 2.5]. Then, by [32, Proposition 2.13], and Lemma 3.1,

$$\begin{aligned} \gamma (T_{{\mathcal {G}}})=\left\| T_{{\mathcal {G}}}^{\dagger }\right\| ^{-1}\ge \sqrt{A}, \end{aligned}$$
(3.3)

and thus

$$\begin{aligned} \lambda _{1}+\frac{\mu }{\gamma (T_{{\mathcal {G}}})\sqrt{1-\delta ^{2}}}\le \lambda _{1}+\frac{\mu }{\sqrt{A(1-\delta ^{2})}}<1. \end{aligned}$$
(3.4)

Collecting (3.2), (3.4) and Lemma 3.2 leads to the fact that \(\mathrm{{range}}(T_{{\mathcal {F}}})\) is closed, and

$$\begin{aligned} \left\| T_{{\mathcal {F}}}^{\dagger }\right\| \le \frac{(1+\lambda _{2})\left\| T_{{\mathcal {G}}}^{\dagger }\right\| }{(1-\lambda _{1})\sqrt{1-\delta ^{2}}-\mu \left\| T_{{\mathcal {G}}}^{\dagger }\right\| }.\ \end{aligned}$$
(3.5)

This implies that \({\mathcal {F}}\) is a HS-frame sequence by [16, Lemma 2.5]. Observe that its optimal lower bound is \(\left\| T_{{\mathcal {F}}}^{\dagger }\right\| ^{-2}\) by [32, Proposition 2.13]. By (3.3) and (3.5),

$$\begin{aligned} \left\| T_{{\mathcal {F}}}^{\dagger }\right\| ^{-2}&\ge \Bigg (\frac{(1-\lambda _{1})\sqrt{1-\delta ^{2}}-\mu \left\| T_{{\mathcal {G}}}^{\dagger }\right\| }{(1+\lambda _{2})\left\| T_{{\mathcal {G}}}^{\dagger }\right\| }\Bigg )^{2}\\&\ge \Bigg (\frac{(1-\lambda _{1})\sqrt{1-\delta ^{2}}-\mu (\sqrt{A})^{-1}}{(1+\lambda _{2})(\sqrt{A})^{-1}}\Bigg )^{2}\\&=\Bigg (\frac{(1-\lambda _{1})\sqrt{A(1-\delta ^{2})}-\mu }{(1+\lambda _{2})}\Bigg )^{2}. \end{aligned}$$

The proof is completed. \(\square \)

The following theorem deals the perturbation of HS-Riesz sequences, it is an application of Theorem 3.1.

Theorem 3.2

Let \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) be a HS-Riesz sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with bounds A and B. Suppose that \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) is a sequence in \({\mathcal {B}}({\mathcal {H}},{\mathcal {C}}_{2})\), and that there exist constants \(\lambda _{2}\in [0,\,1)\) and \(\lambda _{1}\), \(\mu \ge 0\) such that \(\lambda _{1}+\frac{\mu }{\sqrt{A}}<1\) and

$$\begin{aligned} \left\| \sum _{j\in J}({\mathcal {G}}_{j}^{*}-{\mathcal {F}}_{j}^{*}){\mathcal {A}}_{j}\right\| \le \lambda _{1}\left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}\right\| +\lambda _{2}\left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}\right\| +\mu \left( \sum _{j\in J}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}\right) ^{\frac{1}{2}} \end{aligned}$$
(3.6)

for \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0}{\mathcal {C}}_{2}\). Then \({\mathcal {F}}\) is a HS-Riesz sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with bounds \(\left( \frac{(1-\lambda _{1})\sqrt{A}-\mu }{1+\lambda _{2}}\right) ^{2}\) and \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\).

Proof

Since \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) is a HS-Riesz sequence, it is a HS-frame sequence and \(\ker (T_{{\mathcal {G}}})=0\) by [16, Lemma 2.6]. This implies that

$$\begin{aligned} \delta \left( \ker (T_{{\mathcal {G}}}),\ker (T_{{\mathcal {F}}})\right) =0<1. \end{aligned}$$

By Theorem 3.1, \({\mathcal {F}}\) is a HS-frame sequence with bounds \(\left( \frac{(1-\lambda _{1})\sqrt{A}-\mu }{1+\lambda _{2}}\right) ^{2}\) and \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\), and (3.6) holds for all \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\). By [16, Lemma 2.6], we only need to prove \(\ker (T_{{\mathcal {F}}})=0\) to finish the proof. Next we do it. For \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\),

$$\begin{aligned} \left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}\right\|&\ge \left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}\right\| -\left\| \sum _{j\in J}({\mathcal {G}}_{j}^{*}-{\mathcal {F}}_{j}^{*}){\mathcal {A}}_{j}\right\| \\&\ge (1-\lambda _{1})\left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}\right\| -\lambda _{2}\left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}\right\| -\mu \left( \sum _{j\in J}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}\right) ^{\frac{1}{2}}\\&\ge \left( (1-\lambda _{1})\sqrt{A}-\mu \right) \left( \sum _{j\in J}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}\right) ^{\frac{1}{2}}-\lambda _{2}\left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}\right\| . \end{aligned}$$

It follows that

$$\begin{aligned} \Vert T_{{\mathcal {F}}}{\mathcal {A}}\Vert =\left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}\right\| \ge \frac{(1-\lambda _{1})\sqrt{A}-\mu }{1+\lambda _{2}}\left( \sum _{j\in J}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}\right) ^{\frac{1}{2}}{\hbox { }}\left( {\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\right) . \end{aligned}$$

Thus \(\ker (T_{{\mathcal {F}}})=0\). The proof is completed. \(\square \)

Using the gap between the range spaces of synthesis operators, the following theorem gives a perturbation theorem of HS-frame sequences.

Theorem 3.3

Let \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) be a HS-frame sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with bounds A and B, \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) be a sequence in \({\mathcal {B}}({\mathcal {H}},\,{\mathcal {C}}_{2})\). Write \(G=\overline{\mathrm{span}}\{{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}\}_{j\in J}\), \(F=\overline{\mathrm{span}}\{{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}:{\mathcal {A}}_{j}\in {\mathcal {C}}_{2}\}_{j\in J}\). Suppose that \(\delta (F,\,G)<1\) and there exist constants \(\lambda _{2}\in [0,\,1)\) and \(\lambda _{1},\,\mu \ge 0\) such that

$$\begin{aligned} \lambda _{1}+\frac{\mu }{\sqrt{A}}<\sqrt{1-\delta (F,\,G)^{2}} \end{aligned}$$

and

$$\begin{aligned} \left\| \sum _{j\in J}({\mathcal {G}}_{j}^{*}-{\mathcal {F}}_{j}^{*}){\mathcal {A}}_{j}\right\| \le \lambda _{1}\left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}\right\| +\lambda _{2}\left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}\right\| +\mu \left( \sum _{j\in J}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}\right) ^{\frac{1}{2}} \end{aligned}$$
(3.7)

for \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0}{\mathcal {C}}_{2}\). Then \({\mathcal {F}}\) is a HS-frame sequence in \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with bounds \(\left( \frac{(1-\lambda _{1})\sqrt{A}-\mu }{1+\lambda _{2}}\right) ^{2}\) and \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\).

Proof

By the proof of Theorem 3.1, \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) is a HS-Bessel sequence with bound \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\), and (3.7) holds for all \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus {\mathcal {C}}_{2}\). Since \({\mathcal {F}}\) is a HS-Bessel sequence, we can define a bounded operator \(U:{\mathcal {H}}\rightarrow {\mathcal {H}}\) by

$$\begin{aligned} Uh=\sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{\dagger }h+P_{F^{\perp }}P_{G^{\perp }}h{\hbox { }}(h\in {\mathcal {H}}). \end{aligned}$$

We first to show that the operator U is invertible. Since \({\mathcal {G}}\) is a HS-frame sequence, its canonical dual \(\{{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}\}_{j\in J}\) is also a HS-frame sequence with bounds \(B^{-1}\) and \(A^{-1}\). Arbitrarily fix \(h=h_{1}+h_{2}\) with \(h_{1}\in G\), \(h_{2}\in G^{\perp }\). Then

$$\begin{aligned} h_1=\sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}h_1,\,Uh_1=\sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}h_1{\hbox { and }}Uh_2=P_{F^{\perp }}P_{G^{\perp }}h_2. \end{aligned}$$

Also \(\Vert P_{F}P_{G^{\perp }}\Vert =\delta (F,\,G)\) by Lemma 3.3. It follows that

$$\begin{aligned} \Vert h-Uh\Vert \le&\Vert h_1-Uh_1\Vert +\Vert h_2-Uh_2\Vert \nonumber \\ =&\left\| \sum _{j\in J}({\mathcal {G}}_{j}^{*}-{\mathcal {F}}_{j}^{*}){\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}h_1\right\| +\left\| P_{F}P_{G^{\perp }}h_2\right\| \nonumber \\ \le&\lambda _{1}\left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}h_1\right\| +\lambda _{2}\left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}h_1\right\| +\mu \left( \sum _{j\in J}\left\| {\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}h_1\right\| _{2}^{2}\right) ^{\frac{1}{2}}\nonumber \\&\quad +\Vert P_{F}P_{G^{\perp }}\Vert \Vert h_2\Vert \le \lambda _{1}\left\| h_1\right\| +\lambda _{2}\left\| Uh_1\right\| +\frac{\mu }{\sqrt{A}}\left\| h_1\right\| +\delta (F,\,G)\Vert h_2\Vert \nonumber \\ =&\left( \lambda _{1}+\frac{\mu }{\sqrt{A}}\right) \Vert h_1\Vert +\delta (F,\,G)\Vert h_2\Vert +\lambda _{2}\left\| Uh_1\right\| \\ \le&\sqrt{\left( \lambda _{1}+\frac{\mu }{\sqrt{A}}\right) ^{2}+\delta (F,\,G)^{2}}\cdot \sqrt{\Vert h_1\Vert ^{2}+\Vert h_2\Vert ^{2}}+\lambda _{2}\left\| Uh\right\| \nonumber \\ =&\sqrt{\left( \lambda _{1}+\frac{\mu }{\sqrt{A}}\right) ^{2}+\delta (F,\,G)^{2}}\cdot \Vert h\Vert +\lambda _{2}\left\| Uh\right\| \nonumber \end{aligned}$$
(3.8)

for \(h\in {\mathcal {H}}\). Then, by [6, Lemma 1], U is invertible due to \(0\le \left( \lambda _{1}+\frac{\mu }{\sqrt{A}}\right) ^{2}+\delta (F,\,G)^{2}<1\) and \(0\le \lambda _{2}<1\). Next we prove that \({\mathcal {F}}\) has a positive lower HS-frame bound to finish the proof. Arbitrarily fix \({\widetilde{h}}\in F\). Suppose \(U^{-1}{\widetilde{h}}=f_{1}+f_{2}\) with \(f_{1}\in G\) and \(f_{2}\in G^{\perp }\). Then, by the definition of U, we have

$$\begin{aligned} {\widetilde{h}}&=U(U^{-1}{\widetilde{h}})\\&=\sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}(f_{1}+f_{2})+P_{F^{\perp }}P_{G^{\perp }}(f_{1}+f_{2})\\&=\sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}f_{1}+P_{F^{\perp }}P_{G^{\perp }}f_{2}\\&=\sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}f_{1}\\&=Uf_{1}. \end{aligned}$$

This implies that \(U^{-1}{\widetilde{h}}=f_{1}\in G\), \({\widetilde{h}}\in F\). Thus

$$\begin{aligned} \Vert {\widetilde{h}}\Vert ^{4}=&\left| \left\langle UU^{-1}{\widetilde{h}},\,{\widetilde{h}}\right\rangle \right| ^{2}\nonumber \\ =&\left| \left\langle \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}(U^{-1}{\widetilde{h}}),\,{\widetilde{h}}\right\rangle \right| ^{2}\nonumber \\ \le&\left( \sum _{j\in J}\left| \left\langle {\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}(U^{-1}{\widetilde{h}}),\,{\mathcal {F}}_{j}{\widetilde{h}}\right\rangle _{\tau }\right| \right) ^{2}\nonumber \\ \le&\sum _{j\in J}\left\| {\mathcal {G}}_{j}S_{{\mathcal {G}}}^{-1}(U^{-1}{\widetilde{h}})\right\| _{2}^{2}\cdot \sum _{j\in J}\left\| {\mathcal {F}}_{j}{\widetilde{h}}\right\| _{2}^{2}\nonumber \\ \le&A^{-1}\Vert U^{-1}{\widetilde{h}}\Vert ^{2}\sum _{j\in J}\left\| {\mathcal {F}}_{j}{\widetilde{h}}\right\| _{2}^{2}. \end{aligned}$$
(3.9)

Now we estimate \(\Vert U^{-1}{\widetilde{h}}\Vert \) \(({\widetilde{h}}\in F)\). By (3.8), we have that

$$\begin{aligned} \Vert {\widetilde{g}}-U{\widetilde{g}}\Vert \le \left( \lambda _{1}+\frac{\mu }{\sqrt{A}}\right) \Vert {\widetilde{g}}\Vert +\lambda _{2}\Vert U{\widetilde{g}}\Vert {\hbox { }}({\widetilde{g}}\in G). \end{aligned}$$

and thus

$$\begin{aligned} \Vert {\widetilde{g}}\Vert&\le \Vert U{\widetilde{g}}\Vert +\Vert U{\widetilde{g}}-{\widetilde{g}}\Vert \\&\le \Vert U{\widetilde{g}}\Vert +\left( \lambda _{1}+\frac{\mu }{\sqrt{A}}\right) \Vert {\widetilde{g}}\Vert +\lambda _{2}\Vert U{\widetilde{g}}\Vert \\&=(1+\lambda _{2})\Vert U{\widetilde{g}}\Vert +\left( \lambda _{1}+\frac{\mu }{\sqrt{A}}\right) \Vert {\widetilde{g}}\Vert \end{aligned}$$

for \({\widetilde{g}}\in G\). It follows that

$$\begin{aligned} \Vert {\widetilde{g}}\Vert \le \frac{1+\lambda _{2}}{1-\lambda _{1}-\frac{\mu }{\sqrt{A}}}\Vert U{\widetilde{g}}\Vert {\hbox { }}({\widetilde{g}}\in G). \end{aligned}$$
(3.10)

Substituting \({\widetilde{g}}\) for \(U^{-1}{\widetilde{h}}\) in (3.10) gives

$$\begin{aligned} \Vert U^{-1}{\widetilde{h}}\Vert \le \frac{1+\lambda _{2}}{1-\lambda _{1}-\frac{\mu }{\sqrt{A}}}\Vert {\widetilde{h}}\Vert {\hbox { }}({\widetilde{h}}\in F). \end{aligned}$$
(3.11)

Collecting (3.9) and (3.11) leads to

$$\begin{aligned} \left\| {\widetilde{h}}\right\| ^{4}\le A^{-1}\left( \frac{1+\lambda _{2}}{1-\lambda _{1}-\frac{\mu }{\sqrt{A}}}\Vert {\widetilde{h}}\Vert \right) ^{2}\sum _{j\in J}\left\| {\mathcal {F}}_{j}{\widetilde{h}}\right\| _{2}^{2}{\hbox { }}({\widetilde{h}}\in F), \end{aligned}$$

equivalently,

$$\begin{aligned} \sum _{j\in J}\left\| {\mathcal {F}}_{j}{\widetilde{h}}\right\| _{2}^{2}\ge \left( \frac{(1-\lambda _{1})\sqrt{A}-\mu }{1+\lambda _{2}}\right) ^{2}\left\| {\widetilde{h}}\right\| ^{2}{\hbox { }}({\widetilde{h}}\in F). \end{aligned}$$

The proof is completed. \(\square \)

Observe that \(\delta \left( F,\,G\right) =0\) if \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) is a HS-frame in Theorem 3.3. As an immediate consequence, we have the following corollary which appeared in [32, Theorem 4.3 and Corollary 4.5].

Corollary 3.1

Let \({\mathcal {G}}=\{{\mathcal {G}}_{j}\}_{j\in J}\) be a HS-frame (HS-Riesz basis) for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with bounds A and B. Suppose that \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) is a sequence in \({\mathcal {B}}({\mathcal {H}},{\mathcal {C}}_{2})\), and that there exists constants \(\lambda _{2}\in [0,1)\) and \(\lambda _{1}\), \(\mu \ge 0\) such that \(\lambda _{1}+\frac{\mu }{\sqrt{A}}<1\) and

$$\begin{aligned} \left\| \sum _{j\in J}({\mathcal {G}}_{j}^{*}-{\mathcal {F}}_{j}^{*}){\mathcal {A}}_{j}\right\| \le \lambda _{1}\left\| \sum _{j\in J}{\mathcal {G}}_{j}^{*}{\mathcal {A}}_{j}\right\| +\lambda _{2}\left\| \sum _{j\in J}{\mathcal {F}}_{j}^{*}{\mathcal {A}}_{j}\right\| +\mu \left( \sum _{j\in J}\Vert {\mathcal {A}}_{j}\Vert _{2}^{2}\right) ^{\frac{1}{2}} \end{aligned}$$
(3.12)

for \({\mathcal {A}}=\{{\mathcal {A}}_{j}\}_{j\in J}\in \oplus _{0}{\mathcal {C}}_{2}\). Then \({\mathcal {F}}\) is a HS-frame (HS-Riesz basis) for \({\mathcal {H}}\) with respect to \({\mathcal {K}}\) with bounds \(\left( \frac{(1-\lambda _{1})\sqrt{A}-\mu }{1+\lambda _{2}}\right) ^{2}\) and \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\).

4 A Remark

In this section, we show that, using Theorems 2.1, 3.1-3.3, we can recover some results on frames.

Given a sequence \(f=\{f_{j}\}_{j\in J}\) in \({\mathcal {H}}\), we associate it with a HS-sequence \({\mathcal {F}}=\{{\mathcal {F}}_{j}\}_{j\in J}\) in \({\mathcal {B}}({\mathcal {H}},\,{\mathbb {C}})\) by

$$\begin{aligned} {\mathcal {F}}_{j}h=\left\langle h,\,f_{j}\right\rangle {\hbox { }}(h\in {\mathcal {H}}). \end{aligned}$$
(4.1)

Then \({\mathcal {F}}_{j}^{*}c_{j}=c_{j}f_{j}\) for \(j\in J\) and \(c_{j}\in \mathbb {C}\). It follows that f and \({\mathcal {F}}\) has similar frame properties. Specifically, f is a Bessel sequence (frame, frame sequence, Riesz basis, Riesz sequence) for \({\mathcal {H}}\) if and only if \({\mathcal {F}}\) is a HS-Bessel sequence (HS-frame, HS-frame sequence, HS-Riesz basis, HS-Riesz sequence) for \({\mathcal {H}}\) with respect to \(\mathbb {C}\). In this case f and \({\mathcal {F}}\) have the same synthesis operator, i.e., \(T_{f}=T_{{\mathcal {F}}}\), and thus \(\ker (T_{f})=\ker (T_{{\mathcal {F}}})\), \(\mathrm{{range}}(T_{f})=\mathrm{{range}}(T_{{\mathcal {F}}})\).

Let \(\Gamma =(\gamma _{k,\,j})_{k,\,j\in J}\) define an operator in \({\mathcal {B}}(l^{2}(J))\). Applying (2.22) in Theorem 2.1, we have

$$\begin{aligned} \sum _{j\in J}|\langle h,\,f_{j}\rangle |^{2}=\left\| \Gamma \left( T_{g}^{*}h\right) \right\| ^{2}{\hbox { }}(h\in {\mathcal {H}}). \end{aligned}$$

It leads to the following corollary which appeared in [2, Theorem 4].

Corollary 4.1

Let \(g=\{g_{j}\}_{j\in J}\) be a frame for \({\mathcal {H}}\), and \(\Gamma =(\gamma _{k,\,j})_{k,\,j\in J}\) be a bounded linear operator on \(l^{2}(J)\). Define \(f_{j}=\sum \limits _{k\in J}\gamma _{k,\,j}\,g_{k}\) for \(j\in J\). Then \(f=\{f_{j}\}_{j\in J}\) is a frame for \({\mathcal {H}}\) if and only if there exists a constant \(\alpha \) such that

$$\begin{aligned} \Vert \Gamma c\Vert ^{2}\ge \alpha \Vert c\Vert ^{2} {\hbox { }}\left( c\in \mathrm{{range}}(T_{g}^{*})\right) . \end{aligned}$$

Applying Theorems 3.1–3.3, we have the following Corollaries 4.2–4.4 that appeared in [11, Theorem 3.2] and [14, Theorems 3.1 and 3.2].

Corollary 4.2

Let \(g=\{g_{j}\}_{j\in J}\) be a frame sequence in \({\mathcal {H}}\) with bounds A and B. Let \(f=\{f_{j}\}_{j\in J}\) be a sequence in \({\mathcal {H}}\) and suppose that there exist numbers \(\lambda _{2}\in [0,1)\) and \(\lambda _{1}\), \(\mu \ge 0\) such that

$$\begin{aligned} \left\| \sum _{j\in J}c_{j}(g_{j}-f_{j})\right\| \le \lambda _{1}\left\| \sum _{j\in J}c_{j}g_{j}\right\| +\lambda _{2}\left\| \sum _{j\in J}c_{j}f_{j}\right\| +\mu \left( \sum _{j\in J}|c_{j}|^{2}\right) ^{\frac{1}{2}} \end{aligned}$$

for \(c=\{c_{j}\}_{j\in J}\in l_{0}(J)\). Then f is a Bessel sequence with bound \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\).

If furthermore \(\delta =\delta \left( \ker (T_{g}),\,\ker (T_{f})\right) <1\) and \(\lambda _{1}+\frac{\mu }{\sqrt{A(1-\delta ^{2})}}<1\), then f is a frame sequence in \({\mathcal {H}}\) with lower bound \(\left( \frac{(1-\lambda _{1})\sqrt{A(1-\delta ^{2})}-\mu }{1+\lambda _{2}}\right) ^{2}\).

Corollary 4.3

Let \(g=\{g_{j}\}_{j\in J}\) be a Riesz sequence in \({\mathcal {H}}\) with bounds A and B. Let \(f=\{f_{j}\}_{j\in J}\) be a sequence in \({\mathcal {H}}\) and suppose that there exist numbers \(\lambda _{2}\in [0,1)\) and \(\lambda _{1}\), \(\mu \ge 0\) such that \(\lambda _{1}+\frac{\mu }{\sqrt{A}}<1\) and

$$\begin{aligned} \left\| \sum _{j\in J}c_{j}(g_{j}-f_{j})\right\| \le \lambda _{1}\left\| \sum _{j\in J}c_{j}g_{j}\right\| +\lambda _{2}\left\| \sum _{j\in J}c_{j}f_{j}\right\| +\mu \left( \sum _{j\in J}|c_{j}|^{2}\right) ^{\frac{1}{2}} \end{aligned}$$

for \(c=\{c_{j}\}_{j\in J}\in l_{0}(J)\). Then f is a Riesz sequence in \({\mathcal {H}}\) with bounds \(\left( \frac{(1-\lambda _{1})\sqrt{A}-\mu }{1+\lambda _{2}}\right) ^{2}\) and \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\).

Corollary 4.4

Let \(g=\{g_{j}\}_{j\in J}\) be a frame sequence in \({\mathcal {H}}\) with bounds A and B. Let \(f=\{f_{j}\}_{j\in J}\) be a sequence in \({\mathcal {H}}\). Write \(G=\overline{\mathrm{span}}\{g_{j}\}_{j\in J}\), \(F=\overline{\mathrm{span}}\{f_{j}\}_{j\in J}\). Assume that \(\delta (F,\,G)<1\) and there exist constants \(\lambda _{2}\in [0, 1)\) and \(\lambda _{1},\mu \ge 0\) such that

$$\begin{aligned} \lambda _{1}+\frac{\mu }{\sqrt{A}}<\sqrt{1-\delta (F,\,G)^{2}} \end{aligned}$$

and

$$\begin{aligned} \left\| \sum _{j\in J}c_{j}(g_{j}-f_{j})\right\| \le \lambda _{1}\left\| \sum _{j\in J}c_{j}g_{j}\right\| +\lambda _{2}\left\| \sum _{j\in J}c_{j}f_{j}\right\| +\mu \left( \sum _{j\in J}|c_{j}|^{2}\right) ^{\frac{1}{2}} \end{aligned}$$

for \(c=\{c_{j}\}_{j\in J}\in l_{0}(J)\). Then f is a frame sequence in \({\mathcal {H}}\) with bounds \(\left( \frac{(1-\lambda _{1})\sqrt{A}-\mu }{1+\lambda _{2}}\right) ^{2}\) and \(\left( \frac{(1+\lambda _{1})\sqrt{B}+\mu }{1-\lambda _{2}}\right) ^{2}\).