#### Abstract

The concept of g-basis in Hilbert spaces is introduced, which generalizes Schauder basis in Hilbert spaces. Some results about g-bases are proved. In particular, we characterize the g-bases and g-orthonormal bases. And the dual g-bases are also discussed. We also consider the equivalent relations of g-bases and g-orthonormal bases. And the property of g-minimal of g-bases is studied as well. Our results show that, in some cases, g-bases share many useful properties of Schauder bases in Hilbert spaces.

#### 1. Introduction

In 1946, Gabor  introduced a fundamental approach to signal decomposition in terms of elementary signals. In 1952, Duffin and Schaeffer  abstracted Gabor’s method to define frames in Hilbert spaces. Frame was reintroduced by Daubechies et al.  in 1986. Today, frame theory is a central tool in many areas such as characterizing function spaces and signal analysis. We refer to  for an introduction to frame theory and its applications. The following are the standard definitions on frames in Hilbert spaces. A sequence of elements of a Hilbert space is called a frame for if there are constants so that The numbers are called the lower (resp., upper) frame bounds. The frame is a tight frame if and a normalized tight frame if .

In , Sun raised the concept of g-frame as follows, which generalized the concept of frame extensively. A sequence is called a g-frame for with respect to , which is a sequence of closed subspaces of a Hilbert space , if there exist two positive constants and such that for any We simply call a -frame for whenever the space sequence is clear. The tight -frame, normalized tight -frame, -Riesz basis are defined similarly. We call a -frame sequence, if it is a -frame for . We call a -Bessel sequence, if only the right inequality is satisfied. Recently, -frames in Hilbert spaces have been studied intensively; for more details see  and the references therein.

It is well known that frames are generalizations of bases in Hilbert spaces. So it is natural to view -frames as generalizations of the so-called -bases in Hilbert spaces, which will be defined in the following section. And that is the main object which will be studied in this paper. In Section 2, we will give the definitions and lemmas. In Section 3, we characterize the -bases. In Section 4, we discuss the equivalent relations of -bases and -orthonormal bases. In Section 5, we study the property of -minimal of -bases. Throughout this paper, we use to denote the set of all natural numbers, to denote the set of all integer numbers, and to denote the field of complex numbers. The sequence of always means a sequence of closed subspace of some Hilbert space .

#### 2. Definitions and Lemmas

In this section, we introduce the definitions and lemmas which will be needed in this paper.

Definition 2.1. For each Hilbert space sequence , we define the space by With the inner product defined by , it is easy to see that is a Hilbert space.

Definition 2.2. is called g-complete with respect to if .

Definition 2.3. is called g-linearly independent with respect to if , then , where .

Definition 2.4. is called g-minimal with respect to if for any sequence with and any with , one has .

Definition 2.5. and are called g-biorthonormal with respect to , if

Definition 2.6. We say is g-orthonormal basis for with respect to , if it is g-biorthonormal with itself and for any one has

Definition 2.7. We call a g-basis for with respect to if for any there is a unique sequence with such that .

The following result is about pesudoinverse, which plays an important role in some proofs.

Lemma 2.8 (see ). Suppose that is a bounded surjective operator. Then there exists a bounded operator (called the pseudoinverse of T) for which

The following lemmas characterize g-frame sequence and g-Bessel sequence in terms of synthesis operators.

Lemma 2.9 (see ). A sequence is a g-frame sequence for H with respect to if and only if is a well-defined bounded linear operator from into with closed range.

Lemma 2.10 (see ). A sequence is a g-Bessel sequence for H with respect to if and only if is a well-defined bounded linear operator from into .

The following is a simple property about g-basis, which gives a necessary condition for g-basis in terms of g-complete and g-linearly independent.

Lemma 2.11. If is a g-basis for H with respect to , then is g-complete and g-linearly independent with respect to .

Proof. Suppose for each . Then for each , we have . Hence . Therefore . So . So is g-complete. Now suppose . Since and is a g-basis, so for each . Hence is g-linearly independent.

The following remark tells us that g-basis is indeed a generalization of Schauder basis of Hilbert space.

Remark 2.12. If is a Schauder basis of Hilbert space , then it induces a g-basis of with respect to the complex number field , where is defined by . In fact, it is easy to see that for any , so for any , there exists a unique sequence of constants such that .

Definition 2.13. Suppose is a g-Riesz basis of with respect to and is a g-Riesz basis of with respect to . If there is a homomorphism such that for each , then we say that and are equivalent.

Definition 2.14. If is a g-basis of with respect to , then for any , there exists a unique sequence such that and . We define a map , by for each . Then is well defined. We call it the dual sequence of , in case that is also a g-basis, we call it the dual g-basis of .

The following results link g-Riesz basis with g-basis.

Lemma 2.15 (see ). A g-Riesz basis is an exact g-frame. Moreover, it is g-biorthonormal with respect to its dual .

Lemma 2.16. Let . Then the following statements are equivalent.(1)The sequence is a g-Riesz basis for with respect to .(2)The sequence is a g-frame for H with respect to and is g-linearly independent.(3)The sequence is a g-basis and a g-frame with respect to .

Proof. The equivalent between statements and is shown in of . By Lemma 2.11, we know that if is a g-basis, then it is g-linearly independent, so implies . If is a g-frame for , then for every , , where is the canonical dual g-frame of . Hence for every , there exists a sequence , , such that . Since is g-linearly independent, the sequence is unique. Hence is a g-basis for . So implies .

From Lemma 2.16, it is easy to get the following well-known result, which is proved more directly.

Corollary 2.17. Suppose is a g-Riesz basis for , then has a unique dual g-frame.

Proof. It has been shown that every g-frame has a dual g-frame in , so it suffices to show the uniqueness of dual g-frame for g-Riesz bases. Suppose and are dual g-frames of . Then for every , we have . Hence . But is g-linearly independent by Lemma 2.16, so , that is, for each . Thus, for each , which implies that the dual g-frame of is unique.

The following lemma generalizes the similar result in frames to g-frames.

Lemma 2.18. Suppose and are both g-Bessel sequences for with respect to . Then the following statements are equivalent.(1)For any , .(2)For any , .(3)For any , .
Moreover, any of the above statements implies that and are dual g-frames for each other.

Proof. : Since is a g-Bessel sequence, for any . Since is a g-Bessel sequence, the series is convergent by Lemma 2.10. Let . Then for any , we have So , that is, is established.
: Since for any , we have ; hence for any ,
: From the proof of , we know that for any , is convergent. Let , then for any , we have So , that is, is true.
If any one of the three statements is true, then for any x  H, we have where is the bound for the g-Bessel sequence . So which implies that the g-Bessel sequence is a g-frame. Similarly, is also a g-frame. And that they are dual g-frames for each other is obvious by the equality that for any , .

#### 3. Characterizations of -Bases

In this section, we characterized g-bases.

Theorem 3.1. Suppose that is a g-frame sequence with respect to and it is g-linearly independent with respect to . Let . If for any , set , then(1) is a Banach space,(2)when is a g-basis with respect to as well, is a linear bounded and invertible operator, that is, S is a homeomorphism between and .

Proof. Let , then is convergent as . Hence is a convergent sequence, so it is bounded. So . It is obvious that for , , , we have and . If , then for any , , which implies that . Since is g-linearly independent with respect to , we get that , for . Since is arbitrary, so . Thus is a norm on . Suppose is a Cauchy sequence, where . Then For any fixed , we have Now let . Since is a g-frame sequence with respect to , is a well-defined linear bounded operator with closed range by Lemma 2.9. Since is g-linearly independent with respect to , is injective. Hence is surjective. So by Lemma 2.8, there is a bounded operator , the pseudoinverse of , such that , which implies that . Let denote the canonical basis of , then for any . So ; hence So by inequalities (3.3), we get So for any fixed , is a Cauchy sequence. Suppose . From (3.2), we know that, for any , there exists , such that whenever , we have Fix , since , so whenever , Since is convergent. So there exists , such that whenever , we have . So when , we have So is convergent, thus . Let in (3.7), we get that . Hence is a complete normed space, that is, is a Banach space.
If is a g-basis, then it is g-complete and g-linearly independent with respect to by the Lemma 2.11, then the operator not only is well defined but also is one to one and onto. And for any , we have So is bounded operator. Since is a Banach space, by the Open Mapping Theorem, we get that is a homeomorphism.

Theorem 3.2. Suppose is a -basis of with respect to and is its dual sequence. If is also a -frame sequence of with respect to , then(1), let , then ,(2),(3) is a norm on and .

Proof. Let and be as defined in Theorem 3.1. Then for any , . So
Since , . Thus .
It is obvious that is a seminorm. It is sufficient to show that . For any , we have On the other hand,

Theorem 3.3. Suppose is a g-frame with respect to . Then is a g basis with respect to if and only if there exists a constant such that for any , any and , one has

Proof. Suppose is a g-basis with respect to . Then for any , there exists a unique sequence with for each such that . Let . Then by Theorem 3.2, is a norm on and it is equivalent to . So there exists a constant such that, for any , . Hence for any , any , , we choose , then for any , we have
Let . First, we show that . Since is a g-frame, is dense in . It is sufficient to show that is closed. Suppose and . Denote . Then for any and any , we have, for any , Since , so for any , there exists , such that whenever , we have . In the above inequality, let , we get Since is a g-frame sequence, by inequality (3.4), we have that for any and any . So is convergent for each . Suppose . Then Since so converges to , which implies that . Thus is a closed set. Now we will show that is g-linearly independent. Suppose that , where for each . Since for any and any , we have , hence for any . But from inequality (3.4), we have . So for each . Since is arbitrary, for any . Thus is g-linearly independent. So is a g-basis.

#### 4. Equivalent Relations of -Bases

In this section, the equivalent relations of g-bases were discussed.

Theorem 4.1. Suppose that is a g-basis of with respect to , and is a homeomorphism. Then is a g-basis of with respect to .

Proof. For any , . Since is a g-basis of , there exists a unique sequence and for each such that . So . Suppose there is another sequence and for each such that , then . So . But the expansion for is unique, so for each . Hence is a g-basis of with respect to .

Theorem 4.2. Suppose is a -basis of Hilbert space with respect to , is a -basis of Hilbert space with respect to , and , and are dual sequences of and , respectively. If or is a g-basis, then the following statements are equivalent.(1) and are equivalent.(2) is convergent if and only if is convergent, where for each .
Moreover, any one of the above statements implies that both and are g-bases and they are also equivalent.

Proof. : Suppose there is an invertible bounded operator such that for each and is convergent. Then is convergent. So is convergent. Conversely, if is convergent, then is convergent. So is convergent.
: Without loss of generality, suppose is a g-basis of . Then for any , we have , which is convergent in , so is convergent in . Define operator by . Then is well defined and linear. If , that is, , then for each . So . Hence is injective. For any , , which is convergent in . Then is convergent in . Suppose , but we know that and is a g-basis, so for each . Hence , which implies that is surjective. Next, we want to verify that S is bounded.
For any , let be defined by . Then it is obvious that is well defined and linear. Since thus is a bounded operator. It is easy to see that for any , . So for any , there exists , such that whenever , we have that . Since for any , we have , so for any , we have Hence, by the Banach-Steinhaus Theorem, . Since for any , , we have So is bounded. Hence is a bounded invertible operator from onto . Since for any , we have , that is, , so . Hence for any , we have . Since is a g-basis, so for each , which implies that and are equivalent. In the case that any of the two statements is true, then there is an invertible operator such that for each . Hence, for any , So . Thus , but , and is a g-basis of , it follows that for each . By Theorem 4.1, we know that is also a g-basis and and are equivalent.

Theorem 4.3. Suppose is a g-orthonormal basis for with respect to . Then is a g-basis for with respect to and it is self-dual.

Proof. By the definition of g-orthonormal bases, we know that is a normalized tight g-frame. So for any , . Since for any and for any , we have hence for . For and for any , we have so for any , . Thus if , then for any , we have . Thus for any , there is a unique sequence such that for any and . So is a g-basis. It is obvious that is self-dual.

Theorem 4.4. Suppose is a g-orthonormal basis with respect to . Then is convergent if and only if .

Proof. Since is a g-orthonormal basis, by Theorem 4.3, is a g-basis and a g-frame. So is a g-Riesz basis. Thus there exist constants , such that for any integer , we have So is convergent if and only if .

From Theorems 4.2, 4.3, and 4.4, the following corollary is obvious.

Corollary 4.5. Any two g-orthonormal bases are equivalent.

#### 5. The Property of -Minimal of -Bases

In this section, we studied the property of g-minimal of g-bases.

Theorem 5.1. Suppose is a g-frame sequence. Then(1)if is a g-basis, then is g-minimal;(2)if is g-minimal, then is g-linearly independent.

Proof. . Since is a g-basis and it is also a g-frame sequence, it is easy to see that is a g-frame. Hence is a g-Riesz basis by of Lemma 2.16. Suppose is the unique dual g-frame of . By Lemma 2.15, we know that and are g-biorthonormal, that is, , where . For any and any sequence with and , let . Then for any , , but , so . Hence is g-minimal.
. Suppose is g-minimal. If , where for each , then for any . In fact, if there exists such that , then by inequality (3.4), which implies that . Since , , which contradicts with the fact that is -minimal.

Theorem 5.2. Given sequence .(1)If there exists a sequence , such that and are biorthonormal, then is g-minimal.(2)If there exists a unique sequence such that and are biorthonormal, then is -minimal and g-complete.

Proof. The proof of is similar to the proof of of Theorem 5.1, we omit the details. Now we prove : By we know that is g-minimal. So it only needs to show that is g-complete. Suppose that and for any and any . Since , so , which implies that and are biorthonormal, where for each defined by for any . But it is assumed that there exists a unique sequence such that and are biorthonormal, so , hence , which implies that . So is g-complete.

#### Acknowledgment

This work was partially supported by SWUFE's Key Subjects Construction Items Funds of 211 Project.