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
?