#### Abstract

Given the -orthonormal basis for Hilbert space , we characterize the -frames, normalized tight -frames, and -Riesz bases in terms of the -preframe operators. Then we consider the transformations of -frames, normalized tight -frames, and -Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums and -dual frames of -frames by applying the results of characterizations.

#### 1. Introduction

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 .

The concept of frame first appeared in the late 40s and early 50s (see, [1–3]). The development and study of wavelet theory during the last decades also brought new ideas and attentions to frames because of their close connections. There are many related references on this topic, see; [4–8].

In [9], Sun raised the concept of -frame as follows, which generalized the concept of frame extensively. A sequence is called a *-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 and normalized tight -frame 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 [10–14] and the references therein.

Constructing frames and g-frames is an interesting problem in frame theory and it is also useful in applications. In this respect, many mathematicians considered the algebraic operations among frames, which allows us to construct a large number of new frames from existing frames. For more details, see [15–18] and the references therein. In this paper, we will consider the algebraic operations among -frames; in particular, we will consider the direct sums of -frames and general sums of -frames. We also will consider the transforms of -frames and their dual -frames.

#### 2. -Preframe Operators and Transformations of -Frames

In this section, we introduce the -preframe operators and use them to characterize -frames. We also discuss the transformations of -frames. Since any countable infinite sequence can be made as one indexed by natural number , so all infinite sequences in this paper are assumed to be indexed by . For each sequence , we define the space by with the inner product defined by

*Definition 1 (see, [9]). *Let , .(i)If , then we say that is *-complete*.(ii)If is -complete and there are positive constants and such that for any finite subset and , ,
then we say that is a -Riesz basis for with respect to .(iii)We say is a *-orthonormal basis* for with respect to if it satisfies the following:

*Remark 2. *It is obvious that any -frame is -complete and any -orthonormal basis is a normalized tight -frame.

Lemma 3 (see [11]). *, is g-complete if and only if . *

Lemma 4 (see, [11]). *, is a -Bessel sequence for with bound , if and only if the operator
**
defined by is a well-defined bounded operator with . *

Lemma 5. *Let be a -orthonormal basis for with respect to ; then the sequence is a -Bessel sequence with respect to if and only if there is a unique bounded operator such that , for all . *

*Proof. * Since is a -orthonormal basis for , , for any . If is a -Bessel sequence, then the operator
is well-defined and bounded operator by Lemma 5. Also by the definition of -orthonormal basis, it is easy to see that . So
for all , . Hence , which implies that , . Suppose and for any . Then for any , , we have , that is, Since by Lemma 3, , which means that . Thus the operator is unique.

Since , for all , for any , we have
So is a -Bessel sequence with respect to .

*Definition 6. *Let be a -orthonormal basis for , is a -Bessel sequence for with respect to , and the *-preframe operator* associated with is the operator such that for any . *The **-frame operator * associated with is defined by .

*Remark 7. *It is well known that if is a -frame, then is a well-defined positive invertible operator on and is the canonical dual -frame of ; for details see [9]. In this case, for any , we have the Fourier series type decomposition: . In general, there are many -frames for which the formula holds. All such -frames are called alternate dual -frames of .

Lemma 8. *Suppose is a -orthonormal basis for , is a -Bessel sequence for , and and are the -preframe operator and -frame operator associated with , respectively. Then .*

*Proof. *Since for all , for any we have
So .

Lemma 9 (see, [11]). *A sequence is a -frame for if and only if
**
is a well-defined and bounded mapping from onto . *

*Remark 10. *Since is adjoint, the above condition can be replaced with is a well-defined and bounded invertible mapping.

Lemma 11 (see, [19]). *Suppose that is a bounded surjective operator. Then there exists a bounded operator (called the pseudoinverse of ) for which
*

Theorem 12. *Suppose is a -orthonormal basis for , is a -Bessel sequence for , and and are the -preframe operator and -frame operator associated with , respectively. Then*(1)* is a -frame if and only if is onto.*(2)* is a normalized tight -frame if and only if is a coisometry.*(3)* is a -Riesz basis if and only if is invertible.*(4)* is a -orthonormal basis if and only if is unitary. *

* Proof. *(1) By Lemma 9, if is a -frame, then is invertible, but by Lemma 8, so is onto. If is onto, then is a -Bessel sequence by Lemma 5. So we just need to show the existence of lower frame bound. Since is onto, by Lemma 11. Hence , so for any , . So we have
So is a -frame.

(2) Since is a normalized tight -frame if and only if , but by Lemma 8, so if and only is coisometry.

(3) If is a -Riesz basis then is invertible by [11, Corollary 2.18]. Conversely, suppose is invertible. If for any , , since is -complete, then ; hence . It follows that is -complete. Now for any finite set , and , we have
Since
So we have
Hence is a -Riesz basis.

(4) If is a -orthonormal basis, then is unitary by [11, Corollary 2.19]. Conversely, suppose that is unitary. Then for , and , we have
So is a -orthonormal basis.

Lemma 13. *
(1) Suppose that and is onto. Then is onto if and only if is onto.**
(2) Suppose that and is a coisometry. Then is a coisometry if and only if is a coisometry.**
(3) Suppose that and is invertible. Then is invertible if and only if is invertible.**
(4) Suppose that and is a unitary. Then is unitary if and only if is unitary. *

*Proof. *(1) If is onto, then for any , there exists such that , so is onto. Conversely, if is onto, then for any there exists such that . Since is onto, there exists such that . So , which implies that is onto.

(2) If is a coisometry, then . Since is a coisometry, . So , which implies that is a coisometry. Conversely, if is a coisometry, then . Hence is a coisometry.

(3) If is invertible, then , so is invertible. Conversely, if is invertible, since is invertible, then is invertible.

(4) If is unitary, then , so is unitary. Conversely, if is unitary, since is unitary, so is unitary.

By Theorem 12 and Lemma 13, it is easy to get the following results. We leave the details to the readers.

Theorem 14. *Suppose that is a -orthonormal basis for with respect to , . Define the transformation , then:*(1)*It transforms -frames to -frames if and only if is onto.*(2)*Ittransforms normalized tight -frames to normalized tight -frames if and only if is a coisometry.*(3)*Ittransforms -Riesz bases to -Riesz bases if and only if is invertible.*(4)*Ittransforms -orthonormal bases to -orthonormal bases if and only if is unitary.*

*Remark 15. *The above results are established under the constraint that has a -orthonormal basis with respect to . But in general, the statement (1) in the above Theorem is still true.

Theorem 16. *Given . The transformation , transforms -frames to -frames if and only if is onto.*

*Proof. * If transforms -frames to -frames, then is a -frame whenever is a -frame. Suppose that and are the lower and upper frame bounds of and respectively. Then for any , we have
So if , then , which implies that is one to one. If , then is convergent; that is, there exists such that . Hence , so , that is, , which implies that has closed range. So has closed range. Since , is onto.

If is onto and is a -frame for with frame bounds and , then for any , we have
and since by Lemma 11. So
So is a -frame for , which implies that transforms -frames to -frames.

#### 3. Sums of -Frames

In this section, we will consider the direct sums and usual sums of -frames, which generate -frames for direct sum of Hilbert spaces and sum of Hilbert spaces, respectively.

*Definition 17. *Suppose , then is defined by .

*Remark 18. *Since for any scalars and , any , we have
So is linear. Since for any , we have
where . So is bounded. It is easy to see that for any . In fact, for any , we have
So . The finite direct sum of operators have the same meaning.

*Definition 19. *Let , be normalized tight -frames for Hilbert space with respect to , if is a normalized tight -frame for with respect to , then we call them *strongly disjoint.* More generally, if and are -frames with respect to such that is a -frame for with respect to , then they are called *disjoint. *

In general, let be normalized tight -frames for Hilbert space , if is a normalized -frame for , which is the inner direct sum of copies of , then we call them * strongly disjoint*. If the number of sequences are -frames and their inner direct sum is a -frame for the inner direct sum of copies of , then we call them * disjoint*.

*Definition 20. *Let , be normalized tight -frames for Hilbert space , if is a -orthonormal basis of , then we call them *strongly complementary. *

In general, let be normalized tight -frames for Hilbert space , if is a -orthonormal basis for the inner direct sum of copies of , then we call them * strongly complementary*.

In [8], the authors characterize sums of frames by the topological and algebraic properties of , the set of orthogonal projections on . In this section, we always assume that there exists a -orthonormal basis for with respect to and we will characterize all kinds of sums of -frames through the properties of -preframe operators.

Theorem 21. *Suppose , are normalized tight -frames for Hilbert space , and are the -preframe operators associated with and , respectively; then and are strongly disjoint if and only if .*

*Proof. * Since and are strongly disjoint, so is a normalized tight -frame for . Since , are -preframe operators associated with and , respectively, so and are coisometries by Theorem 12 and , for any . So for any , we have
So . Replace by in the above argument; then we get . Hence for any . Therefore .

Suppose ; then for any , we have
So is a normalized tight -frame for ; that is, and are strongly disjoint.

It is easy to see that for any finite number of normalized tight -frames, we have the following result.

Corollary 22. *Suppose are normalized tight -frames for Hilbert space , are -preframe operators associated with , respectively; then are strongly disjoint if and only if , . *

Theorem 23. *Suppose and are -frames for Hilbert space with respect to , and are -preframe operators associated with and , respectively. If , then and are disjoint.*

*Proof. *Since and are -preframe operators associated with -frames and , respectively, so and are onto operators by Theorem 12. So there exist and such that and by Lemma 11. So and . Hence , so . Similarly, . Since , we have
Since
where . Hence . Since
where . Hence
So is a -frame for ; that is, and are disjoint.

Similarly, for any finite number of -frames, we have the following.

Corollary 24. *Suppose are -frames for Hilbert space , are -preframe operators of , respectively; If , then are disjoint. *

Theorem 25. *Let and be normalized tight -frames for , and are -preframe operators associated with and , respectively. If and are strongly complementary, then and .*

*Proof. *If and are strongly complementary, then is a -orthonormal basis for . So they are of course strongly disjoint, therefore by Theorem 21. Since is a -orthonormal basis for , for any and , , we have
But
Since , we have that .

Similarly, for any finite number of normalized tight -frames, we have

Corollary 26. *Suppose are normalized tight -frames for Hilbert space , are -preframe operators associated with , respectively, if are strongly complementary, then , and .*

The above theorems are related with the inner direct sums of -frames or normalized tight -frames. In the following, we will consider the usual sums of -frames or normalized tight -frames.

Theorem 27. *Suppose that and are -frames for Hilbert space , and are -preframe operators associated with and , respectively. If , then is a -frame for . Moreover, if and are normalized tight -frames and , then is a tight -frame for with bounds . *

*Proof. *Since and are -preframe operators associated with -frames and , respectively, so and for any , where is a -orthonormal basis for . Hence for any . To show is a -frame, it is sufficient to show that is onto by Theorem 12. Since , so
Since is invertible, so for any , there exists such that
So is onto.

If and are normalized tight -frames and , then is a -frame and for any , we have
So is a tight -frame with bound .

Similarly, for any finite number of -frames, we have the following.

Corollary 28. *Suppose are -frames for Hilbert space , are -preframe operators associated with , respectively. If , then is a -frame for . Moreover, if are normalized tight -frames for Hilbert space and , then is a tight -frame for with bound .*

The above theorem can be generalized further as follows.

Theorem 29. *Suppose that and are -frames for Hilbert space , and are -preframe operators associated with and , respectively, such that and . If either or is onto, then is a -frame for . *

*Proof. *Since and are -preframe operators associated with and , respectively, and . So
To show is a -frame it is sufficient to show that is onto by Theorem 12. It is well known that is invertible. If is onto, then for any , there exists such that . So for any , there exists such that
It follows that is onto. In the case that is onto, the result can be proved similarly.

Similarly, for any finite number of -frames, we have the following.

Corollary 30. *Suppose are -frames for Hilbert space , and are -preframe operators associated with , respectively, and , , , . If there exists such that is surjective, then is a -frame for .*

#### 4. Dual -Frames of -Frames

In this section, we study the dual -frames by using the properties of -preframe operators.

Theorem 31. *Suppose -frames and are alternate dual -frame pair for Hilbert space , is a coisometry in . Then and are alternate dual -frame pair for . *

* Proof. *Since and are -frames for and is a coisometry, so and are also -frames for by Theorem 14. Since and are alternate dual -frame pair, we have, for any
So
Replacing by and applying , we have
So and are alternate dual -frame pair for .

More general, we have the following result.

Theorem 32. *Suppose and are alternate dual -frame pair for Hilbert space , is a surjective operator in . Then and are alternate dual -frame pair for . Similarly, and are alternate dual -frame pair for . *

*Proof. *Since is onto, by Lemma 11, so . It follows that is onto. Since and are -frames for , so , , and are -frames for by Theorem 14. Since and are alternate dual -frame pair, we have, for any
So
Replaced by and applying , we have
So and are alternate dual -frame pair for . Interchanging and for any , we get that and are alternate dual -frame pair for .

Theorem 33. *Let be a -frame for . Let denote the canonical dual -frame of and denote the the -preframe operator associated with . If is an orthogonal projection with , then is a -frame for and the canonical dual -frame for is .*

*Proof. *It is easy to see that is a -frame for and is a -orthonormal basis for with respect to . We only need to prove the second part. By definition, the canonical dual -frame for is , where . So . Since , so . Since , so . Hence the -preframe operator associated with is as well. So the canonical dual -frame for is .

The following theorem characterizes the alternate dual -frames in terms of -preframe operators.

Theorem 34. *Let and be -frames for Hilbert space , and let and be the -preframe operators associated with and , respectively. Then and are alternate dual -frame pair if and only if or .*

*Proof. *Since and are -preframe operators associated with and , respectively, so and for any . Hence for any ,
Similarly, . It follows that and are alternate dual -frame pair if and only if or .

*Remark 35. *Throughout this paper we assume that there exists -orthonormal basis for Hilbert space . It is well known that any Hilbert space has orthonormal basis. A natural question is that given Hilbert space , and sequence of Hilbert space , Do the -orthonormal bases for with respect to always exist? if not, what is the condition such that the -orthonormal bases exist? These interesting questions will be our next research subjects.

#### Acknowledgment

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