Some new inequalities for continuous fusion frames and fusion pairs

This paper addresses continuous fusion frames and fusion pairs which are extensions of discrete fusion frames and continuous frames. The study of equalities and inequalities for various frames has seen great achievements. In this paper, using operator methods we establish some new inequalities for continuous fusion frames and fusion pairs. Our results extend and improve ones obtained by Balan, Casazza and Găvruţa.

working on efficient algorithms for signal reconstruction, Balan, Casazza, Edidin and Kutyniok in Balan et al. (2005) pointed out the following surprising proposition, and proved it detailedly in Balan et al. (2007).
Proposition 1 (Balan et al. 2005, Theorem 3.2) Let {f i } i∈I be a Parseval frame for H. For every subset J ⊂ I and every f ∈ H, we have Then the study of inequalities related to (1) has interested many mathematicians. The details can be found in Găvruţa (2006); Guo et al. (2016); Li and Sun (2008); Li and Zhu (2012); Zhu and Wu (2010) and references therein. In particular, Kutyniok in 2007 andGăvruţa in 2006 obtained following two propositions: Proposition 2 (Balan et al. 2007, Proposition 4.1) Let {f j } j∈J ⊂ H be a Parseval frame. For any f ∈ H, J 1 ⊂ J, we have Proposition 3 (Găvruţa 2006, Theorem 3.2) Let {f j } j∈J ⊂ H be a frame and {g j } j∈J ⊂ H be an alternate dual frame of {f j } j∈J . then for any f ∈ H, we have Guo, Leng and Li in Guo et al. (2016) generalized Proposition 1 to discrete fusion frames (Guo et al. 2016, Theorem 4). Motivated by above works, in this paper we generalize Proposition 2 and Proposition 3 to continuous fusion frames and fusion pairs. It is worth expecting that our results have potential applications in the frame theory and signal processing. Indeed, our results can be used to recover many results in the literature. For example, Theorem 1 below reduces to Guo et al. (2016), Theorem 8) when the measure is counting measure, and to Proposition 2 if the fusion frame is taken as the usual frame in addition. Similarly, Corollary 3 can be used to recover Proposition 3. This paper is organized as follows. "Preliminaries" section is an auxiliary one. And in this section, we recall some basic notions and properties. In "Equalities and inequalities for continuous fusion frames" section, using the method of operator theory we obtain some important inequalities for continuous fusion frames which are very different from those in the literature. In "Equalities and inequalities for fusion pairs" section, we derive some inequalities of fusion pairs and some bounds estimates. (1) (3)

Preliminaries
This section is an auxiliary one. First we recall some basic notations and notions . The readers can refer to Casazza and Kutyniok (2004), Christensen (2003), Ahmadi (2008, 2010), Rahimi et al. (2006) for details. Let H , K be separable Hilbert spaces, and I a countable index set. We denote by I H the identity operator on H, Ĥ the collection of all closed subspace of H, and L(H, K ) the set of all bounded linear operators from H into K. For a positive measure space (X, µ), we always assume that v : X → [0, ∞) is measurable mapping on X satisfying v(x) � = 0 for a.e. x ∈ X.
Let F be a mapping from X into H. We denote by L 2 (X, F ) the set of all measurable mappings f : X → H such that, for each x ∈ X, and f (x) ∈ F (x), and X �f (x)� 2 dµ(x) < ∞. Then it is a Hilbert space under the following inner product: The numbers A 1 , B 1 are called lower and upper bounds for the frame, respectively.
Definition 2 (Casazza and Kutyniok 2004) Let H be a separable Hilbert space, {w i : i ∈ I} be a family of closed subspace of Hilbert space H, and {v i : i ∈ I} be a family of weight, i.e., v i > 0 for all i ∈ I. The family {(w i , v i ) : i ∈ I} is a fusion frame, if there exist constants 0 < A 2 ≤ B 2 < +∞ such that where π w i is the orthogonal projection onto the subspace w i . The numbers A 2 , B 2 are called lower and upper frame bounds for the fusion frame, respectively.
Definition 3 (Rahimi et al. 2006) Let (X, µ) be a measure space with positive measure µ, and let f : We call A 3 and B 3 the lower and upper continuous frame bounds, respectively. If only the right-hand inequality of (4) is satisfied, we call {f (x) : x ∈ X} a continuous Bes-

Definition 4 (Faroughi and Ahmadi 2008) Let
is the orthogonal projection onto the space F(x). The numbers A, B are called lower and upper frame bounds for the continuous fusion frame, respectively. If only the right-hand inequality of (5) is satisfied, we call (F , v) a continuous Bessel fusion mapping is called a Parsevel continuous fusion frame.
Remark 1 A continuous fusion frame is a generalization of fusion frame. Indeed, when X is countable, and µ is a counting measure, it is exactly a fusion frame.
Let (F , v) be a continuous fusion frame for H. In Faroughi and Ahmadi (2008), the authors defined the continuous fusion frame operator S F : H → H as follows: It is easy to show that S F is a bounded, positive, self-adjoint and invertible operator.
For any X 1 ⊂ X, denote X c 1 = X\X 1 , and we define the following operators: F are positive and self-adjoin operators.
Definition 5 (Faroughi and Ahmadi 2010) Let (F , v) and (G, v) be continuous Bessel fusion mappings on H. We say that F and G is a fusion pair if for any h ∈ H the following holds

Equalities and inequalities for continuous fusion frames
This section is devoted to some inequalities for continuous fusion frames. For this purpose, we first give a simple property of self-adjoint operators.
Lemma 1 Let T ∈ L(H) be a self-adjoint operator and a, b, c ∈ R, U = aT 2 + bT + cI H , then the following statements hold.
Observe that It follows that by Lemma 1. Also observing that S X 1 F − (S X 1 F ) 2 ≥ 0 and we have Again by Lemma 1, we get Since for h ∈ H, we have (7) by (13) and (15). Next we prove (8). Observe that (11) and that S X 1 F − (S X 1 F ) 2 ≥ 0 by (9). It leads to by Lemma 1. For h ∈ H, we have Collecting (18) and (19) leads to (8). The proof is completed. ) is a -tight continuous fusion frame for H. As an immediate consequence of Theorem 3.1, we have Corollary 1 Let (F, v) be a -tight continuous fusion frame for H. Then for X 1 ⊂ X and h ∈ H, we have Next we will give a equality for tight continuous fusion frames. To do so, we first define two operators S 1 F , S 2 F as follows: where (F , v) is a continuous Bessel fusion mapping on H and {a x : Proof We only treat S 1 F , and the other part S 2 F can be treated similarly. For h ∈ H and X 1 ⊂ X, we have where M = sup x∈X |a x | and a x is the conjugate of a x . This implies that S 1 F is well-defined and �S 1 F h� ≤ BM�h�. Therefore, S 1 F is a bounded linear operator. Now let us compute The proof is completed.

Theorem 2 Let (F , v) be a -tight continuous fusion frame for H. Then for h ∈ H and
where a x is the conjugate of a x .
Proof By Proposition 4, S 1 F , S 2 F are well-defined and Since (F , v) is a -tight continuous fusion frame for H, that is −1 S 1 F + −1 S 2 F = I H . Write Q 1 = −1 S 1 F and Q 2 = −1 S 2 F , then and thus Hence for h ∈ H, we get The proof is completed.
As an immediate consequence of Theorem 2 and Lemma 2, we have Corollary 2 Let (F , v) be a -tight continuous fusion frame for H, {a x : x ∈ X} ∈ l ∞ (X) with a x being real. Then for h ∈ H, we have

Equalities and inequalities for fusion pairs
This section focuses on fusion pairs. We begin with the following lemma which can be proved similarly to Proposition 4.
Lemma 3 Let (F , v) and (G, v) be continuous Bessel fusion mappings on H, F and G be a fusion pair, and {a x : x ∈ X} ∈ l ∞ (X). Define the operator T a as follows: then T a is a bounded linear operator, and Theorem 3 Let (F , v) and (G, v) be continuous Bessel fusion mappings on H, F and G be a fusion pair, and {a x : x ∈ X} ∈ l ∞ (X). Then for h ∈ H, we have  (23) and (24). The proof is completed.
Remark 2 Theorem 2 is a special case of Theorem 3.
Theorem 4 Let (F , v) and (G, v) be continuous Bessel fusion mappings on H, F and G be a fusion pair, and {a x : x ∈ X} ∈ l ∞ (X). Then for h ∈ H, we have = �T a h� 2 + �T 1−a h, h� = �T a h, T a h� + �(I H − T a )h, h� = �T a h, T a h� + �h, h� − �T a h, h�. (24)