Abstract
In this paper, we mainly discuss the constructions of some new K-g-frames which differ from the existing methods. Meanwhile, we use the relation between a positive operator and the frame operator of a K-g-frame to yield a new K-g-frame. We also obtain a necessary and sufficient condition to generate a new K-g-frame. In addition, we correct some recent results which were obtained by Huang and Leng. In the end, we give an equivalent characterization to construct some new tight K-g-frames by two given g-Bessel sequences. Our results generalize and improve some remarkable results.
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1 Introduction
A frame as a generalization of an orthonormal basis, appeared first in the late 1940s and early 1950s, provides us with a powerful theoretical tool because of its redundancy and flexibility. Now a frame plays an important role in sampling theory [1], compressed sensing [2] and a number of other fields. We refer the readers to [3,4,5] for an introduction to frame theory and its applications. In [6], Sun proposed the notion of g-frame, which generalized the concept of frame extensively. We know that though many basic properties of g-frame can be shared with frame, not all the properties between them are same. For example, an exact frame is equivalent to a Riesz basis, but an exact g-frame is not equivalent to a g-Riesz basis. We refer the reader to the papers [6,7,8,9,10,11] for more information about g-frames.
Being an extension of frame, the concept of K-frame was introduced by Găvruţa [12], which allows an atomic decomposition of elements in the range of K. In fact, a K-frame is a more general version of frame. There are many differences between a K-frame and a frame. For instance, the sequence \(\{f_{j}\}_{j\in J}\) is a frame for \({\mathcal {H}}\) if and only if \(\{f_{j}\}_{j\in J}\) is a Bessel sequence for \({\mathcal {H}}\) and the corresponding synthesis operator is surjective, but the sequence \(\{f_{j}\}_{j\in J}\) is a K-frame for \({\mathcal {H}}\) if and only if \(\{f_{j}\}_{j\in J}\) is a Bessel sequence for \({\mathcal {H}}\) and the range of K is involved in the range of the corresponding synthesis operator. For more details on K-frames, see references in [12,13,14,15,16,17,18].
Recently, Xiao et al. [19] put forward the notion of K-g-frame, which is more general than g-frame and K-frame in Hilbert spaces. Naturally, K-g-frame attracts many scholars’ attention. Now it has been a hot topic to make full use of various conditions to construct a new K-g-frame (see [20,21,22,23]). Hua and others gave several methods to generate tight K-g-frames and tight g-frames (see [24]). For more details on K-g-frame, readers can consult [19,20,21,22,23,24].
In this paper, we first construct a K-g-frame from a given K-g-frame and a g-Bessel sequence. Next, we adopt a novel way to generate a new K-g-frame from two existing K-g-frames. We also give a necessary and sufficient condition to yield a K-g-frame. Finally, we give an equivalent characterization of constructing tight K-g-frames by two given g-Bessel sequences. We correct the results of Theorem 3.4 and Corollary 3.17 in [21] and Theorem 3.10 in [22]. We also generalize and improve some remarkable results.
Throughout this paper, we will adopt such notations. \({\mathcal {H}}\) is a separable Hilbert space, and \(I_{{\mathcal {H}}}\) is the identity operator for \({\mathcal {H}}\). \({\mathbb {C}}\) is the set of all complex numbers. \(B({\mathcal {H}_1},{\mathcal {H}_2})\) is a collection of all bounded linear operators from \({\mathcal {H}_1}\) to \({\mathcal {H}_2}\) , where \({\mathcal {H}_1}\), \({\mathcal {H}_2}\) are Hilbert spaces, and if \({\mathcal {H}_1}\mathrm{{ = }}{\mathcal {H}_2}\mathrm{{ = }}{\mathcal {H}}\), \(B({\mathcal {H}_1},{\mathcal {H}_2})\) is denoted by \(B({\mathcal {H}})\). Let \(K \in B({\mathcal {H}})\) and \(K\ne 0\), the range and the kernel of K are denoted by R(K) and N(K), respectively. \(\{{\mathcal {V}}_{j}\}_{j\in J}\) is a sequence of closed subspaces of \({\mathcal {H}}\), where J is a finite or countable index set. \(\ell ^{2}(\{{\mathcal {V}}_{j}\}_{j\in J})\) is defined by
with the inner product
It is trivial that \(\ell ^{2}(\{\mathcal {V}_{j}\}_{j\in J})\) is a Hilbert space.
Definition 1.1
A sequence \(\{\Lambda _{j}:\Lambda _{j}\in B({\mathcal {H}},{\mathcal {V}}_{j})\}_{j\in J}\) is called a g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) if there exist two positive constants A and B such that
The constants A and B are called the lower and upper g-frame bounds, respectively. If only the only right inequality of (1.1) holds, \(\{\Lambda _{j}\}_{j\in J}\) is called a g-Bessel sequence for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with bound B.
If \(\{\Lambda _{j}\}_{j\in J}\) is a g-Bessel sequence for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\), we may define the bounded linear operator \(T_{\Lambda }\) by
\(T_{\Lambda }\) is called the synthesis operator. The adjoint operator \(T_{\Lambda }^{*}\) is given as follows:
\(T_{\Lambda }^{*}\) is called the analysis operator. The operator given by
is called the g-frame operator.
Definition 1.2
Let \(K\in B({\mathcal {H}})\). A sequence \(\{f_j\}_{j\in J}\subset {\mathcal {H}}\) is called a K-frame for \({\mathcal {H}}\) if there exist two positive constants A and B such that
The constants A and B are called the lower and upper K-frame bounds, respectively.
Definition 1.3
([19]) Let \(K\in B({\mathcal {H}})\). A sequence \(\{\Lambda _{j}:\Lambda _{j}\in B({\mathcal {H}},{\mathcal {V}}_{j})\}_{j\in J}\) is called a K-g-frame for \({\mathcal {H}}\) with respect to \(\{{\mathcal {V}}_{j}\}_{j\in J}\) if there exist two positive constants A and B such that
The constants A and B are called the lower and upper K-g-frame bounds, respectively.
Definition 1.4
([24]) Let \(K\in B({\mathcal {H}})\). A sequence \(\{\Lambda _{j}:\Lambda _{j}\in B({\mathcal {H}},{\mathcal {V}}_{j})\}_{j\in J}\) is called a tight K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\), if there exists a positive constant A such that
In order to obtain our main results, we need the following lemmas.
Lemma 1.5
Let \({\mathcal {H}}_{1}\) and \({\mathcal {H}}_{2}\) be two Hilbert spaces, and suppose that \(U:{\mathcal {H}_1}\rightarrow {\mathcal {H}_2}\) is a bounded linear operator with closed range R(U). Then, there exists a unique bounded operator \(U^+: {\mathcal {H}_2}\rightarrow {\mathcal {H}_1}\) satisfying
The operator \(U^+\) is called the pseudo-inverse operator of U.
Lemma 1.6
([13]) Suppose that \(U\in B({\mathcal {H}_1},{\mathcal {H}_2})\) is an operator with closed range, then
Lemma 1.7
([19]) The sequence \(\{\Lambda _j\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) if and only if the synthesis operator \(T_{\Lambda }\) is well defined and bounded, and \(R(K)\subset R(T_{\Lambda })\).
Lemma 1.8
([24]) Let \(\{\Lambda _j\}_{j\in J}\) be a g-Bessel sequence for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). Then, \(\{\Lambda _j\}_{j\in J}\) is a tight K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\), if and only if there exists a positive constant A such that \(S_{\Lambda }=AKK^{*}\), where \(S_{\Lambda }\) is the g-frame operator for \(\{\Lambda _j\}_{j\in J}\).
Lemma 1.9
Let \(\{\Lambda _{j}\}_{j\in J}\) and \(\{\Gamma _{j}\}_{j\in J}\) be g-Bessel sequences for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). If \(U_{1},U_{2}\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\), then \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a g-Bessel sequence for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with bound \((\sqrt{B_{1}}\Vert U_{1}\Vert +\sqrt{B_{2}}\Vert U_{2}\Vert )^{2}\).
The proof is easy, we omit it. Later, we will need the following important result from operator theory:
Theorem 1.10
(Douglas’s theorem [25]) Let \(U_{1}\in B({\mathcal {H}}_{1},{\mathcal {H}})\), \(U_{2}\in B({\mathcal {H}}_{2},{\mathcal {H}})\). Then, the following are equivalent:
- \(\mathrm {(1)}\):
-
\(R(U_{1})\subseteq R(U_{2})\);
- \(\mathrm {(2)}\):
-
\(U_{1}U_{1}^{*}\le \alpha ^{2} U_{2}U_{2}^{*}\) for some \(\alpha >0\);
- \(\mathrm {(3)}\):
-
there exists a bounded operator \(X\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\) so that \(U_{1}=U_{2}X\).
Several ways to generate g-frames have been discussed in [9,10,11]. After the notion of K-frame was proposed, there are some references to give a number of construction methods about K-frames (see [14,15,16,17]). Motivated by recent progress in constructions of some new K-g-frames (see [19,20,21,22,23]), we give two different ways to construct new K-g-frames.
Remark 1.11
In [21, Theorem 3.4], the following statement has been formulated: let \(K\in B({\mathcal {H}})\) and \(\{\Lambda _{j}\}_{j\in J}\) be a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with lower and upper bounds A and B, respectively; if \(U\in B({\mathcal {H}})\) has closed range and \(UK=KU\), then \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is a K-g-frame for R(U) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with lower and upper bounds \(A\Vert U^{+}\Vert ^{-2}\) and \(B\Vert U\Vert ^{2}\), respectively. In Example 1.12, we show that this statement is not true in the general case.
Example 1.12
Suppose \({\mathcal {H}}={\mathbb {C}}^{3}\), \(J=\{1,2,3\}\). Let \(\{e_{j}\}_{j\in J}\) be an orthonormal basis of \({\mathcal {H}}\), and \({\mathcal {V}}_{j}=\overline{span}\{e_{j}\}\). Now define \(K\in B({\mathcal {H}})\), \(U\in B({\mathcal {H}})\) and \(\{\Lambda _{j}\}_{j\in J}\) as follows:
Now we show that \(K^{*}f=\langle f,\,e_{1}\rangle e_{3}+\langle f,\,e_{2}\rangle (e_{1}+e_{2}),\,f\in {\mathcal {H}}\). In fact, for any \(f,m\in {\mathcal {H}}\), we have
Thus, for each \(f\in {\mathcal {H}}\), we obtain
This implies that \(\{\Lambda _{j}\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). It is clear that \(U\in B({\mathcal {H}})\) has closed range. For all \(f\in {\mathcal {H}}\), we get
Then, \(UK=KU\).
The adjoint operator of U is \(U^{*}\), \(U^{*}f=\langle f,\,e_{1}-e_{2}\rangle e_{3},\,f\in {\mathcal {H}}\). Indeed, for all \(f,m\in {\mathcal {H}}\), we have
Choosing \(f=e_{2}-e_{1}\in R(U)=\overline{span}\{e_{1}-e_{2}\}\), we get \(\Vert K^{*}f\Vert ^{2}=3\) and
Hence, \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is not a K-g-frame for R(U) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). Furthermore, \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is not a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) and
Remark 1.13
In [22, Theorem 3.10], the following statement has been formulated: let \(\{\Lambda _{j}\}_{j\in J}\) be an atomic system for K, and let \(S_{\Lambda }\) be the frame operator of \(\{\Lambda _{j}\}_{j\in J}\); let U be a positive operator, then \(\{\Lambda _{j}+\Lambda _{j}U\}_{j\in J}\) is an atomic system for K. In Example 1.14, we show that this statement is not true in the general case.
Example 1.14
Suppose \({\mathcal {H}}={\mathbb {C}}^{3}\), \(J=\{1,2,3\}\). Assume that \(\{e_{j}\}_{j\in J}\) is an orthonormal basis of \({\mathcal {H}}\), and \({\mathcal {V}}_{1}={\mathcal {V}}_{2}=\overline{span}\{e_{1}\},{\mathcal {V}}_{3}=\overline{span}\{e_{3}\}\). Now define \(K\in B({\mathcal {H}})\), \(U\in B({\mathcal {H}})\) and \(\{\Lambda _{j}\}_{j\in J}\) as follows:
Now we show that \(K^{*}f=\langle f,\,e_{2}\rangle e_{1},\,f\in {\mathcal {H}}\). In fact, for all \(f,m\in {\mathcal {H}}\), we get
Hence, for every \(f\in {\mathcal {H}}\), we have
Thus, \(\{\Lambda _{j}\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\).
By a simple calculation, we can obtain that U is a self-adjoint operator. Then, for each \(f\in {\mathcal {H}}\), we conclude
Therefore, U is a positive operator.
It is clear that \(\{\Lambda _{j}+\Lambda _{j}U\}_{j\in J}\) is a g-Bessel sequence for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). By a direct computation, we get
Choosing \(f=e_{2}+3e_{3}\in {\mathcal {H}}\), we get \(\Vert K^{*}f\Vert ^{2}=|\langle f,\,e_{2}\rangle |^{2}=1\) and
This proves that \(\{\Lambda _{j}+\Lambda _{j}U\}_{j\in J}\) is not a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). Let \(S_{\Lambda }\) be the frame operator of \(\{\Lambda _{j}\}_{j\in J}\), then for any \(f\in {\mathcal {H}}\), we have
Now we will show that \(US_{\Lambda }\ne S_{\Lambda }U \); indeed, for all \(f\in {\mathcal {H}}\), we obtain
2 Main Results
Theorem 2.1
Let \(K_{1}\in B({\mathcal {H}}_{1})\) and \(K_{2}\in B({\mathcal {H}}_{2})\). Suppose that \(\{\Lambda _{j}\}_{j\in J}\) is a \(K_{1}\)-g-frame and \(\{\Gamma _{j}\}_{j\in J}\) is a g-Bessel sequence for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with the synthesis operators \(T_{\Lambda }\) and \(T_{\Gamma }\), respectively. Assume \(U_{1}, U_{2}\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\) and \(U_{1}T_{\Lambda }T_{\Gamma }^{*}U_{2}^{*}+U_{2}T_{\Gamma }T_{\Lambda }^{*}U_{1}^{*}+U_{2}T_{\Gamma }T_{\Gamma }^{*}U_{2}^{*}\ge 0\). If \(U_{1}\) has closed range, \(U_{1}K_{1}=K_{2}U_{1}\) and \(R(K_{2}^{*})\subset R(U_{1})\), then \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\).
Proof
Let \(\{\Lambda _{j}\}_{j\in J}\) be a \(K_{1}\)-g-frame for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with frame bounds \(A_{1}\) and \(B_{1}\). Let \(\{\Gamma _{j}\}_{j\in J}\) be a g-Bessel sequence for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with g-Bessel bound \(B_{2}\). By Lemma 1.9, we conclude that \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a g-Bessel sequence for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with bound \((\sqrt{B_{1}}\Vert U_{1}\Vert +\sqrt{B_{2}}\Vert U_{2}\Vert )^{2}\).
For each \(g\in {\mathcal {H}}_{2}\), we obtain
Since \(U_{1}\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\) has closed range, and \(U_{1}K_{1}=K_{2}U_{1}\), it is clear that \(K_{1}^{*}U_{1}^{*}=U_{1}^{*}K_{2}^{*}\). According to Lemma 1.6, for any \(g\in {\mathcal {H}}_{2}\), we get
Thus, for every \(g\in {\mathcal {H}}_{2}\), we obtain
So \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). \(\square \)
Corollary 2.2
Let \(K_{1}\in B({\mathcal {H}}_{1})\) and \(K_{2}\in B({\mathcal {H}}_{2})\). Suppose that \(\{\Lambda _{j}\}_{j\in J}\) is a \(K_{1}\)-g-frame for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). If \(U\in B({\mathcal {H}}_{1}, {\mathcal {H}}_{2})\) has closed range, \(UK_{1}=K_{2}U\) and \(R(K_{2}^{*})\subset R(U)\), then \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\).
Corollary 2.3
Let \(K,U\in B({\mathcal {H}})\). Suppose that \(\{\Lambda _{j}\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). If U is a positive operator and \(US_{\Lambda }=S_{\Lambda }U\), where \(S_{\Lambda }\) is the frame operator of \(\{\Lambda _{j}\}_{j\in J}\), then \(\{\Lambda _{j}+\Lambda _{j}U\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\).
Proof
Assume that \(\{\Lambda _{j}\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). Since
from Theorem 2.1, we need only to prove that \(S_{\Lambda }U+US_{\Lambda }+US_{\Lambda }U\ge 0\). According to Proposition 4.33 in [26], we obtain that there exists a unique positive operator C such that \(U=C^{2}\). In addition, since \(US_{\Lambda }=S_{\Lambda }U\), we have \(CS_{\Lambda }=S_{\Lambda }C\). It follows that
for all \(f\in {\mathcal {H}}\). So from Theorem 2.1, Corollary 2.3 holds. \(\square \)
Remark 2.4
By taking \(U_{1}=U\) and \(U_{2}=0\), we obtain Corollary 2.2. Let \({\mathcal {H}}_{1}={\mathcal {H}}_{2}={\mathcal {H}}\) and \(K_{1}=K_{2}=K\) in Corollary 2.2, we can correct Theorem 3.4 in [21]. In counterexample 1.12, \(R(K^{*})\subset R(U)\) may not be true. Hence, the condition \(R(K^{*})\subset R(U)\) is necessary. From Corollary 2.2, we may obtain Corollary 5.32 in [3], Proposition 2.24 in [9] and Theorem 3.3 in [16], and we also correct Proposition 12 in [17]. In counterexample 1.14, the condition \(US_{\Lambda }=S_{\Lambda }U\) is not true. Hence, this condition is necessary in Corollary 2.3. From Corollary 2.3, we may obtain Theorem 3.11 in [16]. From Theorem 2.1, we improve Theorem 3.2 in [5], Theorem 2.4 in [10], Corollary 4.4 in [11], Theorem 2.12 in [14] and Theorem 3.5 in [22].
Theorem 2.5
Let \(K_{1}\in B({\mathcal {H}}_{1})\) be an operator with closed range, suppose that \(\{\Lambda _{j}\}_{j\in J}\) and \(\{\Gamma _{j}\}_{j\in J}\) are \(K_{1}\)-g-frames for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). Assume \(K_{2}\in B({\mathcal {H}}_{2})\), \(U_{1}, U_{2}\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\) and \(U_{1}T_{\Lambda }T_{\Gamma }^{*}U_{2}^{*}+U_{2}T_{\Gamma }T_{\Lambda }^{*}U_{1}^{*}\ge 0\). Then, \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\), if one of the following conditions holds:
- \(\mathrm {(1)}\):
-
\(P=U_{1}+U_{2}\), \(R(P^{*})\subset R(K_{1})\), \(R(K_{2})\subset R(P)\).
- \(\mathrm {(2)}\):
-
\(Q=U_{1}-U_{2}\), \(R(Q^{*})\subset R(K_{1})\), \(R(K_{2})\subset R(Q)\).
Proof
Let \(\{\Lambda _{j}\}_{j\in J}\) be a \(K_{1}\)-g-frame for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with frame bounds \(A_{1}\) and \(B_{1}\). Let \(\{\Gamma _{j}\}_{j\in J}\) be a \(K_{1}\)-g-frame for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with frame bounds \(A_{2}\) and \(B_{2}\). From Lemma 1.9, we obtain that \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a g-Bessel sequence for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with bound \((\sqrt{B_{1}}\Vert U_{1}\Vert +\sqrt{B_{2}}\Vert U_{2}\Vert )^{2}\).
According to the proof of Theorem 2.1, for all \(g\in {\mathcal {H}}_{2}\), we get
Without loss of generality, assume that statement (1) holds; let \(\lambda =\min \{A_{1},A_{2}\}\), by the parallelogram law and Lemma 1.6, for every \(g\in {\mathcal {H}}_{2}\), we obtain
From \(R(K_{2})\subseteq R(P)\), we conclude that there exists \(\alpha >0\) such that \(K_{2}K_{2}^{*}\le \alpha ^{2}PP^{*}\) by Theorem 1.10. It follows that \(\alpha ^{-2}\Vert K_{2}^{*}g\Vert ^{2}\le \Vert P^{*}g\Vert ^{2}\) for all \(g\in {\mathcal {H}}_{2}\). Thus, for each \(g\in {\mathcal {H}}_{2}\), we get
This proves that \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). \(\square \)
Remark 2.6
From Theorem 2.5, we can get Proposition 2.24 in [9], Proposition 3.6 in [15], Theorem 3.5 in [20] and Proposition 3.2 in [23]. It is natural to consider whether the conditions \(R(P^{*})\subset R(K_{1})\) and \(R(Q^{*})\subset R(K_{1})\) are not necessary in Theorem 2.5. Now we give an example to illustrate that the conditions are essential.
Example 2.7
Let \({\mathcal {H}}_{1}={\mathbb {C}}^{3}\) and \(J=\{1, 2, 3\}\). Assume that \(\{e_{j}\}_{j\in J}\) is an orthonormal basis for \({\mathcal {H}}_{1}\) and \({\mathcal {V}}_{j}=\overline{span}\{e_{j}\}\). Let \(\{g_{j}\}_{j=1}^{4}\) be an orthonormal basis for \({\mathcal {H}}_{2}={\mathbb {C}}^{4}\). Now define \(K_{1}\in B({\mathcal {H}}_{1})\), \(K_{2}\in B({\mathcal {H}}_{2})\), \(U_{1},U_{2}\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\) and \(\{\Lambda _{j}\}_{j\in J}\) as follows:
Let \(\Gamma _{j}=\Lambda _{j},\,j=1,2,3.\) Now we prove that \(K_{1}^{*}f=\langle f,\,e_{1}\rangle e_{1}+\langle f,\,e_{2}\rangle e_{3},\,f\in {\mathcal {H}}_{1}\). Indeed, for all \(f,m\in {\mathcal {H}}_{1}\), we have
Hence, for any \(f\in {\mathcal {H}}_{1}\), we get
It follows that \(\{\Lambda _{j}\}_{j\in J}\) and \(\{\Gamma _{j}\}_{j\in J}\) are \(K_{1}\)-g-frame for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). Let \(T_{\Lambda }\) and \(T_{\Gamma }\) be the corresponding synthesis operators of \(\{\Lambda _{j}\}_{j\in J}\) and \(\{\Gamma _{j}\}_{j\in J}\), respectively. Since \(\Gamma _{j}=\Lambda _{j},\,j=1,2,3\), for every \(f\in {\mathcal {H}}_{1}\), we obtain
Now we show that \(U_{1}^{*}g=\langle g,\,g_{3}\rangle e_{2}+\langle g,\,g_{1}\rangle e_{3},\,\,g\in {\mathcal {H}}_{2}\) and \(U_{2}^{*}g=\langle g,\,g_{3}\rangle e_{2},\,g\in {\mathcal {H}}_{2}\). In fact, for all \(f\in {\mathcal {H}}_{1}\) and \(g\in {\mathcal {H}}_{2}\), we obtain
and
By a direct calculation, we can conclude
for all \(g\in {\mathcal {H}}_{2}\). This implies that \(U_{1}T_{\Lambda }T_{\Gamma }^{*}U_{2}^{*}+U_{2}T_{\Gamma }T_{\Lambda }^{*}U_{1}^{*}\ge 0\).
Now we prove that \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a g-Bessel sequence for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). Indeed, for each \(g\in {\mathcal {H}}_{2}\), we get
The adjoint operator of \(K_{2}\) is \(K_{2}^{*}\), \(K_{2}^{*}g=\langle g,\,g_{1}\rangle g_{2},\,\,g\in {\mathcal {H}}_{2}\). In fact, for any \(g,h\in {\mathcal {H}}_{2}\), we obtain
We can choose \(g=g_{1}\in {\mathcal {H}}_{2}\), then we obtain \(\Vert K_{2}^{*}g\Vert ^{2}=|\langle g,\,g_{1}\rangle |^{2}=1\) and
Therefore, \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is not a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). It is obvious that \(K_{1}\in B({\mathcal {H}}_{1})\) has closed range. Let \(P=U_{1}+U_{2}\) and \(Q=U_{1}-U_{2}\), we have \(Pf=2\langle f,\,e_{2}\rangle g_{3}+\langle f,\,e_{3}\rangle g_{1},\,\,f\in {\mathcal {H}}_{1}\) and \(Qf=\langle f,\,e_{3}\rangle g_{1},\,\,f\in {\mathcal {H}}_{1}\). Hence, we get
but
In the following, we offer an equivalent characterization of generating K-g-frames.
Remark 2.8
In [21, Corollary 3.17], it was stated that \(K\in B({\mathcal {H}})\) is an operator with closed range and \(\{\Lambda _{j}\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\); suppose that \(U\in B({\mathcal {H}})\) has closed range and \(UK=KU\), then the following conditions are equivalent: \(\mathrm {(1)}\) U is surjective; \(\mathrm {(2)}\) \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). We announce a counterexample in Example 2.9.
Example 2.9
Assume \({\mathcal {H}}={\mathbb {C}}^{3}\), \(J=\{1,2,3\}\). Let \(\{e_{j}\}_{j\in J}\) be an orthonormal basis of \({\mathcal {H}}\), and \({\mathcal {V}}_{j}=\overline{span}\{e_{j}\}\). Now define \(K\in B({\mathcal {H}})\), \(U\in B({\mathcal {H}})\) and \(\{\Lambda _{j}\}_{j\in J}\) as follows:
It is clear that \(K\in B({\mathcal {H}})\) has closed range. The adjoint operator of K is \(K^{*}\), \(K^{*}f=\langle f,\,e_{1}\rangle e_{2}+\langle f,\,e_{2}\rangle e_{1},\,f\in {\mathcal {H}}\). In fact, for all \(f,m\in {\mathcal {H}}\), we have
For every \(f\in {\mathcal {H}}\), we get
Thus, \(\{\Lambda _{j}\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\).
It is obvious that \(U\in B({\mathcal {H}})\) has closed range. By a direct calculation, for each \(f\in {\mathcal {H}}\), we obtain
then \(UK=KU\).
By a simple computation, we get \(U^{*}f=\sum _{j=1}^{2}\langle f,\,e_{j}\rangle e_{j},\,\,f\in {\mathcal {H}}\), then for all \(f\in {\mathcal {H}}\), we have
Hence, \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is a K-g-frame for \({\mathcal {H}}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) and it is clear that U is not surjective.
Theorem 2.10
Let \(K_{1}\in B({\mathcal {H}}_{1})\) and \(\{\Lambda _{j}\}_{j\in J}\) be a \(K_{1}\)-g-frame for \({\mathcal {H}}_{1}\) with respect to \(\{{\mathcal {V}}_{j}\}_{j\in J}\). Suppose that \(K_{2}\in B({\mathcal {H}}_{2})\) is an operator with dense range, \(U\in B({\mathcal {H}}_{1}, {\mathcal {H}}_{2})\) is an operator with closed range and \(UK_{1}=K_{2}U\), then \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) if and only if U is surjective.
Proof
Suppose that U is surjective, it is obvious that \(R(K_{2}^{*})\subset R(U)\). According to Corollary 2.2, \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\).
On the other hand, let \(T_{\Lambda }\) be the synthesis operator of the \(K_{1}\)-g-frame \(\{\Lambda _{j}\}_{j\in J}\) and L be the synthesis operator of the \(K_{2}\)-g-frame \(\{\Lambda _{j}U^{*}\}_{j\in J}\), then for any \(\{g_{j}\}_{j\in J}\in {\ell ^{2}}(\{\mathcal {V}_{j}\}_{j\in J})\), we obtain
Hence, we get \(L=UT_{\Lambda }\).
Since \(\{\Lambda _{j}U^{*}\}_{j\in J}\) is a \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\), we have \(R(K_{2})\subset R(L)\) by Lemma 1.7. Hence \(R(K_{2})\subset R(UT_{\Lambda })\subset R(U)\). Then, we obtain \(\overline{R(K_{2})}\subset \overline{R(U)}\). Since \(K_{2}\in B({\mathcal {H}}_{2})\) is an operator with dense range and \(U\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\) is an operator with closed range, then U is surjective. \(\square \)
Remark 2.11
By taking \({\mathcal {H}}_{1}={\mathcal {H}}_{2}={\mathcal {H}}\) and \(K_{1}=K_{2}=K\), we may correct Corollary 3.17 in [21]. In counterexample 2.9, we obtain that statement (2) does not imply statement (1).
In the end, we give a necessary and sufficient condition to yield a series of tight K-g-frames by two existing g-Bessel sequences.
Theorem 2.12
Suppose that \(\{\Lambda _{j}\}_{j\in J}\) and \(\{\Gamma _{j}\}_{j\in J}\) are g-Bessel sequences for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with synthesis operators \(T_{\Lambda }\), \(T_{\Gamma }\) and frame operators \(S_{\Lambda }\), \(S_{\Gamma }\), respectively. Let \(K\in B({\mathcal {H}}_{2})\) and \(U_{1}, U_{2}\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\). Then, \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a tight K-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) if and only if there exists \(A>0\) such that
Proof
According to Lemma 1.9, we get that \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a g-Bessel sequence for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\). Let L be the synthesis operator of \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\), then for each \(\{g_{j}\}_{j\in J}\in {\ell ^{2}}(\{\mathcal {V}_{j}\}_{j\in J})\), we obtain
Thus \(L=U_{1}T_{\Lambda }+U_{2}T_{\Gamma }\). Suppose that S is the frame operator of \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\), then
By Lemma 1.8, \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a tight K-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) if and only if there exists \(A>0\) such that
This completes the proof. \(\square \)
From Lemma 1.8 and Theorem 2.12, we have the following corollary.
Corollary 2.13
Let \(K_{1}\in B({\mathcal {H}}_{1})\), \(\{\Lambda _{j}\}_{j\in J}\) and \(\{\Gamma _{j}\}_{j\in J}\) be tight \(K_{1}\)-g-frames for \({\mathcal {H}}_{1}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) with frame bounds \(A_{1}\) and \(A_{2}\). Let \(K_{2}\in B({\mathcal {H}}_{2})\) and \(U_{1}, U_{2}\in B({\mathcal {H}}_{1},{\mathcal {H}}_{2})\). Assume that \(T_{\Lambda }\) and \(T_{\Gamma }\) are synthesis operators of \(\{\Lambda _{j}\}_{j\in J}\) and \(\{\Gamma _{j}\}_{j\in J}\), respectively. Then, \(\{\Lambda _{j}U_{1}^{*}+\Gamma _{j}U_{2}^{*}\}_{j\in J}\) is a tight \(K_{2}\)-g-frame for \({\mathcal {H}}_{2}\) with respect to \(\{\mathcal {V}_{j}\}_{j\in J}\) if and only if there exists \(A>0\) such that
Remark 2.14
We can obtain Theorem 2.1 in [18] and Theorem 10 in [24] from Theorem 2.12. From Corollary 2.13, we may obtain Theorem 2.2, Theorem 2.3, Theorem 2.4 and Theorem 2.5 in [18]; meanwhile, we can also get Theorem 14, Theorem 16, Theorem 18 and Theorem 20 in [24]. We improve Theorem 2.7 in [10] and Theorem 4.7 in [11] by Corollary 2.13.
References
Poon, C.: A consistent and stable approach to generalized sampling. J. Fourier Anal. Appl. 20, 985–1019 (2014)
Tsiligianni, E.V., Kondi, L.P., Katsaggelos, A.K.: Construction of incoherent unit norm tight frames with application to compressed sensing. IEEE Trans. Inf. Theory 60(4), 2319–2330 (2014)
Christensen, O.: An Introduction to Frames and Riesz Bases, 2nd edn. Birkhäuser, Boston (2016)
Casazza, P.G.: The art of frame theory. Taiwan. J. Math. 4(2), 129–201 (2000)
Obeidat, S., Samarah, S., Casazza, P.G., Tremain, J.C.: Sums of Hilbert space frames. J. Math. Anal. Appl. 351, 579–585 (2009)
Sun, W.C.: G-frames and g-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)
Zhu, Y.C.: Characterization of g-frames and g-Riesz bases in Hilbert spaces. Acta Math. Sin. Engl. Ser. 24(10), 1727–1736 (2008)
Li, J.Z., Zhu, Y.C.: Exact g-frames in Hilbert spaces. J. Math. Anal. Appl. 374, 201–209 (2011)
Najati, A., Faroughi, M.H., Rahimi, A.: G-frames and stability of g-frames in Hilbert spaces. Methods Funct. Anal. Topol. 14(3), 271–286 (2008)
Li, D.W., Leng, J.S., Huang, T.Z., Sun, G.M.: On sums and stability of g-frames in Hilbert spaces. Linear Multilinear Algebra 66(8), 1578–1592 (2018)
Guo, X.X.: Characterizations of disjointness of g-frames and constructions of g-frames in Hilbert spaces. Complex Anal. Oper. Theory 8, 1547–1563 (2014)
Găvruţa, L.: Frames for operators. Appl. Comput. Harmon. Anal. 32, 139–144 (2012)
Zhu, Y.C., Shu, Z.B., Xiao, X.C.: \(K\)-frames and \(K\)-Riesz bases in complex Hilbert spaces (in Chinese). Sci. Sin. Math. 48(5), 609–622 (2018)
Guo, X.X.: Canonical dual \(K\)-Bessel sequences and dual \(K\)-Bessel generators for unitary systems of Hilbert spaces. J. Math. Anal. Appl. 444, 598–609 (2016)
Xiao, X.C., Zhu, Y.C., Găvruţa, L.: Some properties of \(K\)-frames in Hilbert spaces. Results Math. 63, 1243–1255 (2013)
Jia, M., Zhu, Y.C.: Some results about the operator perturbation of a \(K\)-frame. Results Math. 73(4), Articles 138 (2018)
He, M., Leng, J.S., Yu, J.L., Li, D.W.: Probability modelled optimal \(K\)-frame for erasures. IEEE. Access 6, 54507–54515 (2018)
Ding, M.L., Xiao, X.C., Zeng, X.M.: Tight \(K\)-frames in Hilbert spaces (in Chinese). Acta Math. Sin. 56(1), 105–112 (2013)
Xiao, X.C., Zhu, Y.C., Shu, Z.B., Ding, M.L.: G-frame with bounded linear operator. Rocky Mt. J. Math. 45(2), 675–693 (2015)
Hua, D.L., Huang, Y.D.: \(K\)-g-frames and stability of \(K\)-g-frames in Hilbert spaces. J. Korean Math. Soc. 53(6), 1331–1345 (2016)
Huang, Y.D., Shi, S.N.: New constructions of \(K\)-g-frames. Results Math. 73(4), Articles 162 (2018)
Li, D.W., Leng, J.S., Huang, T.Z.: Generalized frames for operators associated with atomic systems. Banach J. Math. Anal. 12(1), 206–221 (2018)
Dastourian, B., Janfada, M.: G-frames for operators in Hilbert spaces. Sahand. Commun. Math. Anal. 8(1), 1–21 (2017)
Huang, Y.D., Hua,D.L.: Tight \(K\)-g-frame and its novel characterizations via atomic systems. Adv. Math. Phys. Article 3783456 (2016). https://doi.org/10.1155/2016/3783456
Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Am. Math. Soc. 17(2), 413–415 (1966)
Douglas, R.G.: Banach Algebra Techniques in Operator Theory, 2nd edn. Academic Press, New York (1998)
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Du, D., Zhu, YC. Constructions of K-g-Frames and Tight K-g-Frames in Hilbert Spaces. Bull. Malays. Math. Sci. Soc. 43, 4107–4122 (2020). https://doi.org/10.1007/s40840-020-00911-0
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DOI: https://doi.org/10.1007/s40840-020-00911-0