#### Abstract

-frame is a generalization of -frame. We generalize the tight -frame to --frame via atomic systems. In this paper, the definition of tight --frame is put forward; equivalent characterizations and necessary conditions of tight --frame are given. In particular, the necessary and sufficient condition for tight --frame being tight -frame is obtained. Finally, by means of methods and techniques of frame theory, several properties of tight --frame are given.

#### 1. Introduction

Frame in Hilbert space was first introduced in 1952 by Duffin and Schaeffer [1] to deal with nonharmonic Fourier series and reintroduced in 1986 by Daubechies et al. [2]. Since then the frame theory began to be more wildly studied. Today, frame theory has been widely used in filter theory [3], image processing [4], numerical analysis, and other areas. We refer to [5–8] for an introduction to frame theory in Hilbert space and its application.

With the deepening of research on frame theory, various generalizations of frames have been proposed; see [9–12]. Atomic systems for subspaces were first introduced by Feichtinger and Werther in [13] based on examples arising in sampling theory. In 2011, Găvruţa [14] introduced -frame in Hilbert spaces to study atomic decomposition systems and discussed some properties of them. In [15–18], some conclusions of -frame were given. With the extensive research of -frame and -frame in Hilbert space, Zhu et al. [19, 20] began to study --frame, which was limited to the range of a bounded linear operator in Hilbert space and had gained greater flexibility in practical application relative to -frame. --frame, as a more general frame than -frame and -frame, has become one of the most active fields in frame theory in recent years. In [20, 21], several properties and characterizations of --frame were obtained. However, many problems of --frame have not been studied. Based on these important results of --frame, we extend tight -frame to --frame and put forward the concept of tight --frame. In this paper, we give equivalent characterizations and necessary conditions of tight --frame for Hilbert space. We also obtain the necessary and sufficient condition of tight --frame to be tight -frame. Finally, we present several properties of tight --frame for Hilbert space.

Throughout this paper, is separable Hilbert space and is the identity operator. is a collection of all bounded linear operators from to , where and are two Hilbert spaces. In particular, is a collection of all bounded linear operators from to . For any , is the range of and is the adjoint operator of . is a sequence of closed subspaces of , where is a subset of integers . is defined by with the inner product given by It is clear that is a complex Hilbert space.

#### 2. Preliminaries

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

*Definition 1 (see [9, Definition 1.1]). *A sequence is called a -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 -frame, respectively. If the right inequality is satisfied, then is said to be a -Bessel sequence for with respect to . If , we call this -frame a tight -frame, and if , it is called a Parseval -frame.

For a -Bessel sequence , defines a bounded linear operator, that is, The adjoint operator is given by By composing with its adjoint , we obtain the bounded linear operator We call , , and the preframe operator, analysis operator, and frame operator of -Bessel sequence, respectively.

*Definition 2 (see [22, Definition 2.6]). *We say is -orthonormal basis for with respect to , if it is -biorthonormal with itself, that is, , , , , and for any one has .

*Definition 3 (see [21, Theorem 2.5]). *Let and for any . A sequence is called a --frame for with respect to if there exist constants such that The constants and are called the lower and upper bounds of --frame, respectively.

*Remark 4. *Every --frame is a -Bessel sequence for with respect to . If , then --frame is just the ordinary -frame.

Motivated by the definition of tight -frame, we give the following definition of tight --frame.

*Definition 5. *Let and for any . A sequence is called a tight --frame for with respect to if there exists constant such that The constant is called the bound of tight --frame. If , we call this tight --frame a Parseval --frame for with respect to .

*Remark 6. *If , then tight --frame and Parseval --frame are tight -frame and Parseval -frame, respectively.

*Definition 7. *Let . An operator is said to be left-invertible if there exists an operator such that . The operator is called a left-inverse of ; that is, . Similarly, an operator is said to be right-invertible if there exists an operator such that . The operator is called a right-inverse of ; that is, . If there exists such that , then we say that is invertible and is an inverse of ; that is, .

Lemma 8 (see [23, Theorem 1]). *Let and . The following conditions are equivalent:*(1)*.*(2)*There exists such that .*(3)*There exists a bounded operator so that .*

Lemma 9 (see [21, Theorem 2.5]). *Let . Then the following statements are equivalent:*(1)* is a --frame for with respect to .*(2)* is a -Bessel sequence for with respect to and there exists a -Bessel sequence for with respect to such that *

#### 3. Properties of Tight --Frame for Hilbert Space

In this section, we first give characterizations of tight --frame and then give several properties of tight --frame.

Theorem 10. *Let ; , and let be the preframe operator of . Then the following statements are equivalent:*(1)* is a tight --frame for with respect to with bound .*(2)*There exists constant such that for any .*(3)*There exists constant such that .*

*Proof. *. Suppose that is a tight --frame for with respect to with bound . By the definition of tight --frame, we get Since is the preframe operator of , we have This implies that for any .

. If there exists constant such that for any , then we obtain Hence .

. If there exists constant such that , then , . That is, Therefore, is a tight --frame for with respect to with bound . The proof of Theorem 10 is completed.

Corollary 11. *Suppose that is a tight --frame for with respect to with bound , is the preframe operator of , and is the frame operator of ; then*(1)*;*(2)*;*(3)*.*

*Proof. *Theorem 10 together with Lemma 8 shows that (1) and (2) are satisfied. We only need to prove that (3) holds. Assume that is a tight --frame for with respect to with bound . By Theorem 10, we have , . Therefore,The proof of Corollary 11 is completed.

Lemma 9 gives an equivalent characterization of --frame; does the tight --frame have the similar characterization? Clearly, if is a tight --frame for with respect to , then there exists a -Bessel sequence for with respect to such that for any . The theorem below gives a necessary condition of tight --frame.

Theorem 12. *Let . Suppose that is a tight --frame for with respect to with bound . Then there exists a -Bessel sequence for with respect to with bound such that , , and .*

*Proof. *Let be a tight --frame for with respect to . By Lemma 9, there exists a -Bessel sequence for with respect to such that , . For any , we have Via (15), Since is a tight --frame for with respect to with bound , we get Furthermore, Therefore, . The proof of Theorem 12 is completed.

Note that when , tight --frame is tight -frame. One may wonder whether when tight --frame is tight -frame as well. In fact, the answer is negative. The following example demonstrates this.

*Example 13. *Suppose that ; . Let be an orthonormal basis of , and let . Now define and as follows:By a simple calculation, we haveFor any , we have Obviously, and are Parseval -frames for with respect to .

Define the bounded linear operator as follows: Now we prove that is a tight --frame for with respect to . For any , we have Hence, . It follows that Therefore, for any , we have . Via the definition of tight --frame, is a tight --frame for with respect to . For any , we have .

Example 13 shows that if a tight --frame is a tight -frame, then cannot be . In the following theorem, we state a necessary and sufficient condition for a tight --frame being a tight -frame.

Theorem 14. *Let and . Suppose that is a tight --frame for with respect to with bound . Then is a tight -frame for with respect to with bound if and only if is right-invertible and the right-invertible operator is .*

*Proof. *First, we prove the sufficient condition. Since is a tight --frame for with respect to with bound , we have Assume that is a tight -frame for with respect to with bound . Then, for any , we get By (25) and (26), for any , implying that . Then, for any , we have . This implies that . So is right-invertible and the right-invertible operator is .

Next, we prove the necessary condition. Suppose that is right-invertible and the right-invertible operator is . Then ; that is, . So That is, Since is a tight --frame for with respect to with bound , we have This implies that is a tight -frame for with respect to with bound . The proof of Theorem 14 is completed.

In the following, we will verify whether the in Example 13 is equal to . For any , we have It follows that . This implies that is right-invertible and the right-invertible operator is .

Corollary 15. *Let . Suppose that is a Parseval --frame for with respect to ; then is a Parseval -frame for with respect to if and only if is right-invertible and the right-invertible operator is .*

To enrich the theory of atomic systems, we give several important properties of tight --frame in the following.

Theorem 16. *Let . If is a tight -frame for with respect to with bound , then is a tight --frame for with respect to with bound .*

*Proof. *Since is a tight -frame for with respect to with bound , we have Again, for any , we have ; then Therefore, is a tight --frame for with respect to with bound . The proof of Theorem 16 is completed.

Corollary 17. *Let . If is a -orthonormal basis for with respect to , then is a tight --frame for with respect to .*

Theorem 18. *Let . If is a tight --frame for with respect to with bound , then is a tight --frame for with respect to with bound .*

*Proof. *Since is a tight --frame for with respect to with bound , we have And, for any , we have ; then So is a tight --frame for with respect to with bound . The proof of Theorem 18 is completed.

Corollary 19. *Let . If is a tight --frame for with respect to , then is a tight --frame for with respect to , where is a given positive integer.*

Theorem 20. *Let , and let be the collection of all tight --frames for with respect to . Then if and only if there exists such that .*

*Proof. *. If is a -orthonormal basis for with respect to , by Corollary 17, we get that is a tight --frame for with respect to . Since , we have that is a tight --frame for with respect to . Assume that the bound of tight --frame is . By the definition of tight --frame, we obtain Since is a -orthonormal basis for with respect to , we have for any . By the definition of -orthonormal basis, we get By (35) and (36), we get for any . So .

. Suppose that is a tight --frame for with respect to with bound ; then Under this assumption, there exists such that . So for any , we have . Hence Therefore, is a tight --frame for with respect to with bound . The proof of Theorem 20 is completed.

Corollary 21. *Let and let be the collection of all Parseval --frames for with respect to . Then if and only if .*

#### 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) and Natural Science Foundation of Ningxia (Grant no. NZ13084).