A New Inequality for Frames in Hilbert Spaces

A frame for a Hilbert space firstly emerged in the work on nonharmonic Fourier series owing to Duffin and Schaeffer [1], which has made great contributions to various fields because of its nice properties; the reader can examine the papers [2–12] for background and details of frames. Balan et al. in [13] showed us a surprising inequality when they further investigated the Parseval frame identity derived from their study on efficient algorithms for signal reconstruction, which was then extended to general frames and alternate dual frames by Găvruţa [14]. In this paper, we establish a new inequality for frames in Hilbert spaces, where a scalar and a bounded linear operator with respect to two Bessel sequences are involved, and it is shown that our result can lead to the corresponding results of Balan et al. and Găvruţa. The notations H, IdH, and J are reserved, respectively, for a complex Hilbert space, the identity operator onH, and an index set which is finite or countable. The algebra of all bounded linear operators onH is designated as B(H). One says that a family {fj}j∈J of vectors inH is a frame, if there are two positive constants C,D > 0 satisfying


Introduction
A frame for a Hilbert space firstly emerged in the work on nonharmonic Fourier series owing to Duffin and Schaeffer [1], which has made great contributions to various fields because of its nice properties; the reader can examine the papers [2][3][4][5][6][7][8][9][10][11][12] for background and details of frames.
Balan et al. in [13] showed us a surprising inequality when they further investigated the Parseval frame identity derived from their study on efficient algorithms for signal reconstruction, which was then extended to general frames and alternate dual frames by Gȃvruţa [14]. In this paper, we establish a new inequality for frames in Hilbert spaces, where a scalar and a bounded linear operator with respect to two Bessel sequences are involved, and it is shown that our result can lead to the corresponding results of Balan et al. and Gȃvruţa.
The notations H, Id H , and J are reserved, respectively, for a complex Hilbert space, the identity operator on H, and an index set which is finite or countable. The algebra of all bounded linear operators on H is designated as (H).
One says that a family { } ∈J of vectors in H is a frame, if there are two positive constants , > 0 satisfying The frame { } ∈J is said to be Parseval if = = 1. If { } ∈J satisfies the inequality to the right in (1), we call that { } ∈J is a Bessel sequence for H.
For a given frame F = { } ∈J , the frame operator F , a positive, self-adjoint, and invertible operator on H, is defined by from which we see that where the involved frame {̃= −1 F } ∈J is said to be the canonical dual of { } ∈J .
For any I ⊂ J, denote I = J \ I. A positive, bounded linear, and self-adjoint operator induced by I and the frame F = { } ∈J is given below

The Main Results
We need the following simple result on operators to present our main result. Lemma 1. Suppose that , , ∈ (H) and that + = .
en for each ∈ [0, 1] we have * + ( * + * ) From this fact and taking into account that we arrive at the relation stated in the lemma.
We can immediately get the following result obtained by Poria in [15], when putting = Id H in Lemma 1.

Journal of Function Spaces
A similar discussion yields We also have Thus the result follows from Theorem 3.
Let { } ∈J be a Parseval frame for H; then F = Id H . Thus for any I ⊂ J, Similarly we have This together with Corollary 4 leads to a result as follows.
Proof. Since { } ∈J is an alternate dual frame of { } ∈J , On the one hand we have On the other hand we have By Theorem 3 the conclusion follows.
As a matter of fact, we can establish a more general inequality for alternate dual frames than that shown in Corollary 6.
This completes the proof.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that he has no conflicts of interest.