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Journal of Function Spaces
Volume 2015, Article ID 465094, 9 pages
http://dx.doi.org/10.1155/2015/465094
Research Article

The Characterization and Stability of g-Riesz Frames for Super Hilbert Space

School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China

Received 12 July 2015; Accepted 10 September 2015

Academic Editor: Hans G. Feichtinger

Copyright © 2015 Dingli Hua and Yongdong Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

G-frames and g-Riesz frames as generalized frames in Hilbert spaces have been studied by many authors in recent years. The super Hilbert space has a certain advantage compared with the Hilbert space in the field of studying quantum mechanics. In this paper, for super Hilbert space , the definitions of a g-Riesz frame and minimal g-complete are put forward; also a characterization of g-Riesz frames is obtained. In particular, we generalize them to general super Hilbert space . Finally, a conclusion of the stability of a g-Riesz frame for the super Hilbert space is given.

1. Introduction

Frame as generalized basis in Hilbert space was first introduced by Duffin and Schaeffer [1] during the studied nonharmonic Fourier series in 1952. In 1986, Daubechies et al. [2] reintroduced the concept of frame. Now the theory of frames has been widely used in many areas such as the characterization of function spaces, signal processing, filter theory, image processing and quantum mechanics. We refer to [310] for an introduction to frame theory in Hilbert space and its application.

Sun [11] introduced g-Riesz basis and g-frame; g-frame actually generalized the concept of frame. Since then, g-frame, g-frame sequence, Besselian g-frame, near g-Riesz bases, and so on are focused on and studied by many authors. The authors [12] introduced Besselian g-frames and near g-Riesz bases in Hilbert space and gave some characterizations of them. In [13], g-Riesz frames were studied and some corresponding results were given. In [14], the concept of g-bases in Hilbert spaces was introduced and some properties about g-bases were proved. Because super Hilbert spaces arose naturally as the state space of a quantum field in the functional Schrödinger representation of spinor quantum field theory and it provided a means to bring supersymmetric quantum field theories into a form resembling standard quantum mechanics, the super Hilbert space has certain advantages compared with the Hilbert space in quantum mechanics. With the extensive research of super Hilbert space [1520], scholars were beginning to study g-frames for super Hilbert spaces [20, 21]. Unfortunately, although g-Riesz frames were considered as a class of important frames, we have not consulted the literature of the g-Riesz frame for super Hilbert space so far. Because g-Riesz frames play an important role in approximate calculating coefficients of g-frames, therefore, the study of g-Riesz frames for super Hilbert space has a double meaning of theory and application. In order to enrich the frame theory, we give the concept of g-Riesz frame for super Hilbert space and the characterization and necessary condition for g-Riesz frame. We also expand corresponding conclusions to general super Hilbert space . Finally, we consider the stability of a g-Riesz frame for super Hilbert space.

Throughout this paper, and are two complex Hilbert spaces and is a sequence of closed subspaces of . is the collection of all bounded linear operators from into , where is a subset of integers . is defined by with the inner product given by and it is clear that is a complex Hilbert space.

The literature [16] gave the definition of super Hilbert space.

Definition 1 (see [16, p.557]). Super Hilbert space is a direct sum that of two complex Hilbert spaces equipped with the super Hermitian form .

2. Preliminaries

In this section, some necessary definitions and lemmas are introduced.

Definition 2 (see [11, Definition  1.1]). A sequence is called a g-frame for with respect to if there exist two positive constants and such that, for all , The constants and are called the lower and upper bounds of g-frame, respectively. If the right hand inequality holds, then we say that is a g-Bessel sequence for with respect to . If , we call this g-frame a tight g-frame. If a g-frame ceases to be a g-frame whenever any single element is removed from , it is called an exact g-frame.

Definition 3 (see [14, Definition  2.2]). One says that is g-complete, if .

Definition 4 (see [11, Definition  3.1]). A sequence is called a g-Riesz basis for with respect to , if the sequence is g-complete and there exist positive constants and such that for all finite subset and , . The constants and are called the lower and upper bounds of g-Riesz bases, respectively.

Definition 5 (see [11, Definition  3.1]). Let . Suppose that is an orthonormal basis for , where is a subset of . Denote We call the sequence induced by with respect to .

Theorem 6 (see [21, Proposition  2.8]). Let and be sequences in and , respectively, and let be an orthonormal basis for , where is a subset of , and let , , and . Then is a frame (resp., Bessel sequence, Riesz basis) for super Hilbert space if and only if is a g-frame (resp., g-Bessel sequence, g-Riesz basis) for with respect to .

Proposition 7 (see [21, Proposition  2.9]). Let , , and be g-frames, where . Then g-frame operator for is defined by

The literature [21] introduced the concept of disjoint g-frames. A pair of g-frames and is called disjoint if is a g-frame for , where .

With the definition of g-complete of g-frame for Hilbert space, we give the definition of g-complete of g-frame for super Hilbert space as follows.

Definition 8. is called g-complete with respect to under the condition of that .

3. Characterization of g-Riesz Frame for Super Hilbert Space

In this section, we first give the concept and the characterization of g-Riesz frame for super Hilbert space , and then we generalize them to super Hilbert space .

3.1. Characterization of g-Riesz Frame for Super Hilbert Space

Before giving the characterization of g-Riesz frames for super Hilbert space , we give the definition of g-Riesz frames and some related lemmas.

Suppose that is a subset of , and denote

Definition 9. Suppose that is a g-frame for with respect to . One says that is a g-Riesz frame if every subfamily of is a g-frame for with respect to with uniform g-frame lower bounds.

For the above , we have the following.

Lemma 10. Suppose that for every , , , , and is an orthonormal basis for . and are defined as in (5). Then .

Proof. DenoteSince , , , , we have , , . This implies that . Therefore, .
On the other hand, suppose that , and then there exists a finite subset and , , such that . For every , let , where . Then, we have Therefore, This implies that . The proof of Lemma 10 is completed.

Lemma 11. Let be a closed subspace of and . Suppose that for every , , where is a subset of and , are defined as in (5). If is g-complete in , then is complete in .

Proof. For any , , we have , where is a closed subspace of . Furthermore, we obtain It is enough to prove that if and , then for , . By equality (6), we obtain Since is g-complete in , we have . Hence, is complete in .

Inspired by the concept of minimal g-complete of g-frame for Hilbert space, we give the definition of minimal g-complete of g-frame for super Hilbert space.

Definition 12. Let be a closed subspace of and be a subset of . If is g-complete in , but is not g-complete in for any , then one says that is minimal g-complete in .

Lemma 13. Suppose that for and is any finite nonempty subset of . Then there exists a finite nonempty subset such that is minimal g-complete in and , where is defined as in (8).

Proof. We prove Lemma 13 in two cases.
Case  1. is g-complete in for any , but is not g-complete in . Let . Then the conclusion is right.
Case  2. Suppose that there exists such that is g-complete in . Let . If there still exists such that is complete in , then we remove from in the same way. Repeat the operation above. Because is nonempty and finite, this process must stop after finite steps. Assume that we remove from , where and . Then, we obtain that satisfies the following two statements:
is g-complete in .
For any , is not g-complete in .
By Lemmas 10 and 11, the statement implies that
is complete in .
Obviously, is nonempty. The proof is by contradiction.
Suppose , by , we obtain that is g-complete in . It is obvious that this is impossible. So is nonempty. Now we prove . By Definition 12 and Lemma 10, is minimal g-complete in . By , we get By Lemma 10, we get . The proof of Lemma 13 is completed.

Based on this, we can obtain a characterization of g-Riesz frame for super Hilbert space .

Theorem 14. Let be a g-frame for with respect to . Then the following two statements are equivalent.
(1) is a g-Riesz frame for with respect to .
(2) There exists such that is minimal g-complete in for any nonempty subset of . And where is defined as in (8).

Proof. . Since is a g-Riesz frame for with respect to , there exists such that is a g-frame for with respect to for any nonempty of . Then, we obtain . Suppose that is a g-frame for with respect to with upper bound . Then There are two cases to prove that is a g-frame for with respect to , where is any nonempty subset of .
Case  1. When is any finite nonempty subset of , by Lemma 13, there exists a finite nonempty subset such that is minimal g-complete in and . By (15), we have Again by (15), for , we have Case  2. Let be any infinite subset of and . Then, for any , there exists a finite subset and , such that . Denote Clearly, . Suppose that there exists and such that . By , we get . Let , we get , where . By Case  1, we have Now we prove . We only need to prove that for every . From (6), we obtain By Lemma 10, we have for any , . From (22), we get Again by (6), we have Using (21), we get Letting , for any . The proof of Theorem 14 is completed.

3.2. Characterization of g-Riesz Frame for Super Hilbert Space

Definition 15. Let be a g-frame for with respect to . If any subsequence is also a g-frame for with respect to with uniform g-frame lower bound, then one says that is a g-Riesz frame for with respect to .
Like (8), we can define as follows: where is any finite subset of and , . Similarly, we give two lemmas.

Lemma 16. For every , let be an orthonormal basis of and , where , . Then

Lemma 17. Let be a closed subspace of and let be an orthonormal basis of . Suppose that is a subset of and . For every , , one has , where . If is g-complete in , then is complete in .

In terms of the concept of minimal g-complete of g-frame for Hilbert space, we give the definition of minimal g-complete of g-frame for super Hilbert space .

Definition 18. Let be a closed subspace of and . If is g-complete in , but is not g-complete in for any , and then one says that is minimal g-complete in .

Lemma 19. Suppose that and is any finite nonempty subset of . Then there exists a finite nonempty subset such that is minimal g-complete in and where is defined as (26).

Proof. The proof is similar to proof of Lemma 10.

By above lemma, we can get a characterization of g-Riesz frame for super Hilbert space . Obviously, Theorem 14 is the special case of Theorem 20.

Theorem 20. Let be a g-frame for with respect to . Then the following two statements are equivalent. is a g-Riesz frame for with respect to .For any nonempty subset of , there exists , if is minimal g-complete in . And

Proof. The proof is analogous to proof of Theorem 14.

4. Stability of g-Riesz Frames for Super Hilbert Space

In this section, we use the characterization of g-Riesz frame for super Hilbert space in Section 3 to study the stability of g-Riesz frame for super Hilbert space .

The stability of g-frames is important in practice which is wildly studied by many authors; for example, see [2224]. The following is a fundamental result in the study of the stability of g-frames.

Proposition 21 (see [23, Theorem  3.1]). Suppose that is a g-frame for with respect to with bounds and . There exists such that . If satisfies for , then is a g-frame for with respect to with bounds

Example 22 illustrates a g-Riesz frame of super Hilbert space has no result of stability like Proposition 21.

Example 22. Suppose that is an orthonormal basis of , where and . Let . Define the bounded linear operator as follows:

First, we prove that is a g-frame for with respect to . In fact, for any , we have Thus, is a g-frame for with respect to .

For any , let , and we have It implies that . Since is an orthonormal basis for , we have So .

Next, we prove that is a g-Riesz frame for with respect to . Let be any finite subset of . From Lemma 10, we obtain that Hence, for every , we have . Therefore, is minimal g-complete in . Now we prove that is minimal g-complete in .

If there exists such that , then we have Since is an orthonormal basis for , we have and . From the minimal complete of for in and , we can obtain . This implies that is minimal g-complete in . Then, for any , we have By Theorem 14, is a g-Riesz frame for with respect to .

Let . Define the bounded linear operator as follows: By direct calculation, for any , suppose that , and we have . Let , . Then

Now we prove that is not a g-Riesz frame for with respect to .

The proof is by contradiction. Suppose that is a g-Riesz frame for with respect to . Let . By Lemma 10, we have Choosing , then Since is a g-frame for with respect to , there exists such that Letting , it implies that . But this is a contradiction. We conclude that is not a g-Riesz frame for with respect to .

On the other hand, for any , we have

From Example 22, we can realize that , . Suppose that is a g-Riesz frame for with respect to . Even if and satisfy the inequality of Proposition 21, it is uncertain to get that is a g-Riesz frame for with respect to .

Theorem 23. Let be a g-Riesz frame for with respect to with bounds and , . If satisfies for any , where is any finite subset of , then is a g-Riesz frame for with respect to with bounds

Proof. Since is a g-frame for with respect to with bounds and , for any finite subset , then By the triangle inequality, we have From (45), (47), and (48), we have Therefore, the series is convergent. By (45), we have By Proposition 21, we get that is a g-frame for with respect to . For any finite subset , by the triangle inequality, we obtain Using (45) and (51), we have For any finite subset , denote If is minimal g-complete in , we need to prove that ; that is, . For any , since , there exist and such that . From Lemma 10, we obtain that and , where , for , . Therefore, for any finite , we have By (45), we have .
Then for any , it follows that . Since is g-complete in , we have . Therefore , and it implies that .
For any , since is a g-frame for with respect to with lower bound , by (52), we have By Theorem 14, we get that is a g-Riesz frame for with respect to . The proof of Theorem 23 is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grants nos. 61261043 and 10961001), Natural Science Foundation of Ningxia (Grant no. NZ13084), and Innovation Project for Graduate of Beifang University of Nationalities.

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