Some equalities and inequalities for fusion frames

Motivated by the work of Balan et al. (2007]) and G?vru?a (2006]), in this section, we continue this work about fusion frames and get some important
equalities and inequalities of these frames in a different case.

Lemma 1

(Zhu and Wu 2010]) Let P and Q be two linear bounded operators on such that. Then.

Now, we present main theorems of this section.

Theorem 4

Let be a fusion frame for with the fusion frame operator S, is the alternate dual fusion frame of. Then, for any and any,

Proof

For any , we define a bounded linear operator as

Clearly, . This, together with Lemma 1, implies that

(7)

In the situation of Parseval fusion frames the equality is of special form.?

Corollary 1

Let be a Parseval fusion frame for with the fusion frame operator, is the alternate dual fusion frame of. Then, for any and any,

Remark 2

Clearly, when the dual fusion frame of is itself, i.e., , which was obtained Theorem 2.2 in Xiyan et al. (2009]) as a particular case from the above result.

In fact, similarly to the proof of Theorem 4, we can give a more general result as
follow. Moreover, the result has another proof in Xiao et al. (2014]).

Theorem 5

Let be a fusion frame for with the fusion frame operator S, is the alternate dual fusion frame of. Then, for any and any,

(8)

where is the conjugata of .

Remark 3

Let be a tight fusion frame for with the fusion frame bound A, and is real for any . In this case, using the Theorem 5, we obtain

Lemma 2

(G?vru?a 2006]) Let P and Q are two self-adjoint bounded linear operators in and. Then we have

Theorem 6

Let be a fusion frame for with the fusion frame operator S, is the dual fusion frame of. Then, for any and any, we have

Proof

Applying , we have that . Combining this with Lemma 2, it follows that

(9)

Replacing f by , one has

Combining this with and , the proof is completed.?

Remark 4

The identity of above was established Theorem 2.1 in Xiyan et al. (2009]), but the inequality in this form is a new result.

Corollary 2

Let be a tight fusion frame for with the fusion frame bound A. Then

In addition, if is a Parseval fusion frame for, then we have

(10)

Proof

Since be a tight fusion frame for with the fusion frame bound A, then for any ,

and

It follows from Theorem 6 that, for any ,

?

Theorem 7

Let be a tight fusion frame for with the fusion frame bound A. Then, for any with, and any, we have

(11)

Proof

Applying Corollary 2 yields that

Similarly Corollary 3.6 in Xiao and Zeng (2010]), obtain

Corollary 3

Let be a tight fusion frame for with the fusion frame bound A. Then, for any, where is a positive integer, with, for, . Then for any, we have

where are positive integers satisfying .

Proof

Applying (11), replace J and E by and , the above result hold.?

The inequality (10) in Corollary 2 leads us to introduce some notations and . Let be a Parseval fusion frame. For any and , define

and

Theorem 8

and have the following properties:

1.
;

2.
, ;

3.
, .

Proof

By inequality (10), holds trivially.

For any and any , we have

Hence,

This implies that . That is .

(2) and (3) follow directly by inequality (10) in Corollary 2.?

Some results for the Parseval fusion frame were established in Xiyan et al. (2009]). For the reader’s convenience and our results equivalence, we not only recall its
formulation but also provide its proof as follows.

Theorem 9

Let be a Parseval fusion frame for. Then, for any and any, the following statements are equivalent:

1.
;

2.
;

3.
;

4.
.

Proof

(1) (2). Since is a Parseval fusion frame, then for any , we have . This implies that

Applying (10), (3) (2) (1) hold trivially.

(2) (4) follows from

?