# A New Inequality for Frames in Hilbert Spaces

Authors: Zhong-Qi Xiang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We obtain a new inequality for frames in Hilbert spaces associated with a scalar and a bounded linear operator induced by two Bessel sequences. It turns out that the corresponding results due to Balan et al. and Găvruţa can be deduced from our result.


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## Body

## 1. 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 [2–12] 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 notationsH, IdH, 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 B(H).One says that a family{fj}j∈J of vectors in H is a frame, if there are two positive constants C,D>0 satisfying(1)Cf2≤∑j∈Jf,fj2≤Df2,∀f∈H.The frame {fj}j∈J is said to be Parseval if C=D=1. If {fj}j∈J satisfies the inequality to the right in (1), we call that {fj}j∈J is a Bessel sequence for H.For a given frameF={fj}j∈J, the frame operator SF, a positive, self-adjoint, and invertible operator on H, is defined by (2)SF:H→H,SFf=∑j∈Jf,fjfj,∀f∈H, from which we see that (3)f=∑j∈Jf,fjSF-1fj=∑j∈Jf,SF-1fjfj,∀f∈H, where the involved frame f~j=SF-1fjj∈J is said to be the canonical dual of {fj}j∈J.For anyI⊂J, denote Ic=J∖I. A positive, bounded linear, and self-adjoint operator induced by I and the frame F={fj}j∈J is given below (4)SIF:H→H,SIFf=∑j∈If,fjfj,∀f∈H.Suppose thatF={fj}j∈J and G={gj}j∈J are two Bessel sequences for H. An application of the Cauchy-Schwartz inequality can show that the operator(5)SFG:H→H,SFGf=∑j∈Jf,gjfjis well-defined and further SFG∈B(H). Particularly, if SFG=IdH, then both {fj}j∈J and {gj}j∈J are frames for H. In this case we say that {gj}j∈J is an alternate dual frame of {fj}j∈J, and the pair ({fj}j∈J,{gj}j∈J) is called an alternate dual frame pair.


## 2. The Main Results

We need the following simple result on operators to present our main result.Lemma 1.
Suppose thatU,V,L∈B(H) and that U+V=L. Then for each λ∈[0,1] we have (6)U∗U+λV∗L+L∗V=V∗V+1-λU∗L+L∗U+2λ-1L∗L≥2λ-λ2L∗L.Proof.
A direct calculation gives(7)U∗U+λV∗L+L∗V=U∗U+λL∗-U∗L+L∗L-U=U∗U+λL∗L-U∗L+L∗L-L∗U=U∗U-λU∗L+L∗U+2λL∗L.From this fact and taking into account that (8)V∗V+1-λU∗L+L∗U+2λ-1L∗L=L∗-U∗L-U+1-λU∗L+L∗U+2λ-1L∗L=L∗L-L∗U+U∗L+U∗U+1-λU∗L+L∗U+2λ-1L∗L=U∗U-λU∗L+L∗U+2λL∗L=U-λL∗U-λL+2λ-λ2L∗L≥2λ-λ2L∗L,we arrive at the relation stated in the lemma.We can immediately get the following result obtained by Poria in [15], when putting L=IdH in Lemma 1.Corollary 2.
Suppose thatU,V∈B(H) and that U+V=IdH. Then for every λ∈[0,1] we have (9)U∗U+λV∗+V=V∗V+1-λU∗+U+2λ-1IdH≥2λ-λ2IdH.Theorem 3.
Suppose that{fj}j∈J is a frame for H, that {gj}j∈J and {hj}j∈J are two Bessel sequences for H, and that the operator SFG is defined by (5). Then for each λ∈[0,1] and each f∈H, we have(10)∑j∈Jf,gj-hjfj2+Re⁡∑j∈Jf,hjfj,SFGf=∑j∈Jf,hjfj2+Re⁡∑j∈Jf,gj-hjfj,SFGf≥2λ-λ2Re⁡∑j∈Jf,gj-hjfj,SFGf+1-λ2Re⁡∑j∈Jf,hjfj,SFGf.Moreover, if SFG is self-adjoint, then for any λ∈[0,1] and any f∈H,(11)∑j∈Jf,fjgj-hj2+Re⁡∑j∈Jf,SFGhjfj,f=∑j∈Jf,fjhj2+Re⁡∑j∈Jf,SFGgj-hjfj,f≥2λ-λ2Re⁡∑j∈Jf,SFGgj-hjfj,f+1-λ2Re⁡∑j∈Jf,SFGhjfj,f.Proof.
We takeUf=∑j∈Jf,gj-hjfj and Vf=∑j∈Jf,hjfj for any f∈H. Then U,V∈B(H) and further (12)Uf+Vf=∑j∈Jf,gj-hjfj+∑j∈Jf,hjfj=∑j∈Jf,gjfj=SFGf.By Lemma 1 we have (13)Uf2+2λRe⁡SFG∗Vf,f=Vf2+21-λRe⁡SFG∗Uf,f+2λ-1Re⁡SFGf,SFGf.Therefore, (14)Uf2=Vf2+21-λRe⁡SFG∗Uf,f+2λ-1Re⁡SFGf,SFGf-2λRe⁡SFG∗Vf,f=Vf2+2Re⁡SFG∗Uf,f-2λRe⁡SFG∗U+SFG∗Vf,f+2λ-1Re⁡SFGf,SFGf=Vf2+2Re⁡SFG∗Uf,f-Re⁡SFGf,SFGf=Vf2+2Re⁡SFG∗Uf,f-Re⁡SFG∗U+Vf,f=Vf2+Re⁡SFG∗Uf,f-Re⁡SFG∗Vf,f. It follows that (15)∑j∈Jf,gj-hjfj2+Re⁡∑j∈Jf,hjfj,SFGf=Uf2+Re⁡SFG∗Vf,f=Vf2+Re⁡SFG∗Uf,f=∑j∈Jf,hjfj2+Re⁡∑j∈Jf,gj-hjfj,SFGf. We now prove the inequality in (10). Again by Lemma 1, (16)Uf2+2λRe⁡SFG∗Vf,f≥2λ-λ2SFG∗SFGf,f for every f∈H. Hence, (17)Uf2≥2λ-λ2SFG∗SFGf,f-2λRe⁡SFG∗Vf,f=2λ-λ2SFGf,SFGf-2λRe⁡Vf,SFGf=2λ-λ2Re⁡U+Vf,SFGf-2λRe⁡Vf,SFGf=2λ-λ2Re⁡Uf,SFGf-λ2Re⁡Vf,SFGf, from which we conclude that (18)∑j∈Jf,gj-hjfj2+Re⁡∑j∈Jf,hjfj,SFGf=Uf2+Re⁡Vf,SFGf≥2λ-λ2Re⁡Uf,SFGf+1-λ2Re⁡Vf,SFGf=2λ-λ2Re⁡∑j∈Jf,gj-hjfj,SFGf+1-λ2Re⁡∑j∈Jf,hjfj,SFGf.
The proof of (11) is similar to the proof of (10); we leave the details to the reader.Corollary 4.
Suppose that{fj}j∈J is a frame for H with frame operator SF and that f~j=SF-1fj for any j∈J. Then for all λ∈[0,1], for any I⊂J and any f∈H, we have (19)∑j∈Icf,fj2+∑j∈JSIFf,f~j2=∑j∈If,fj2+∑j∈JSIcFf,f~j2≥2λ-λ2∑j∈If,fj2+1-λ2∑j∈Icf,fj2.Proof.
Settinggj=SF-1/2fj for each j∈J, then SFG=SF1/2. Taking (20)hj=0,j∈I,gj,j∈Ic, then {gj}j∈J and {hj}j∈J are both Bessel sequences for H. For any f∈H we have (21)∑j∈Jf,fjgj-hj2=∑j∈If,fjSF-1/2fj2=SF-1/2SIFf2=SF-1/2SIFf,SF-1/2SIFf=SIFf,SF-1SIFf=SFSF-1SIFf,SF-1SIFf=∑j∈JSF-1SIFf,fjfj,SF-1SIFf=∑j∈JSIFf,SF-1fjSF-1fj,SIFf=∑j∈JSIFf,f~j2. A similar discussion yields (22)∑j∈Jf,fjhj2=∑j∈JSIcFf,f~j2. We also have (23)Re⁡∑j∈Jf,SFGhjfj,f=∑j∈Icf,fj2,Re⁡∑j∈Jf,SFGgj-hjfj,f=∑j∈If,fj2. Thus the result follows from Theorem 3.Let{fj}j∈J be a Parseval frame for H; then SF=IdH. Thus for any I⊂J, (24)∑j∈JSIFf,f~j2=∑j∈JSIFf,fj2=SIFf2=∑j∈If,fjfj2,∀f∈H. Similarly we have (25)∑j∈JSIcFf,f~j2=∑j∈Icf,fjfj2. This together with Corollary 4 leads to a result as follows.Corollary 5.
Suppose that{fj}j∈J is a Parseval frame for H. Then for each λ∈[0,1], for any I⊂J and any f∈H, we have (26)∑j∈Icf,fj2+∑j∈If,fjfj2=∑j∈If,fj2+∑j∈Icf,fjfj2≥2λ-λ2∑j∈If,fj2+1-λ2∑j∈Icf,fj2.Corollary 6.
Suppose that({fj}j∈J,{gj}j∈J) is an alternate dual frame pair for H. Then for each λ∈[0,1], for any I⊂J and any f∈H, we have (27)∑j∈If,gjfj2+Re⁡∑j∈Icf,gjfj,f=∑j∈Icf,gjfj2+Re⁡∑j∈If,gjfj,f≥2λ-λ2Re⁡∑j∈If,gjfj,f+1-λ2Re⁡∑j∈Icf,gjfj,f.Proof.
Since{gj}j∈J is an alternate dual frame of {fj}j∈J, SFG=IdH. For any j∈J, let (28)hj=0,j∈I,gj,j∈Ic. On the one hand we have (29)∑j∈Jf,gj-hjfj2=∑j∈If,gjfj2,∑j∈Jf,hjfj2=∑j∈Icf,gjfj2. On the other hand we have (30)Re⁡∑j∈Jf,hjfj,SFGf=Re⁡∑j∈Icf,gjfj,f,Re⁡∑j∈Jf,gj-hjfj,SFGf=Re⁡∑j∈If,gjfj,f. By Theorem 3 the conclusion follows.Remark 7.
Theorems 2.2 and 3.2 in [14] and Proposition 4.1 in [13] can be obtained when taking λ=1/2, respectively, in Corollaries 4, 6, and 5.As a matter of fact, we can establish a more general inequality for alternate dual frames than that shown in Corollary6.Theorem 8.
Suppose that({fj}j∈J,{gj}j∈J) is an alternate dual frame pair for H. Then for every bounded sequence {ωj}j∈J, for all λ∈[0,1] and all f∈H, we have (31)Re⁡∑j∈Jωjf,gjfj,f+∑j∈J1-ωjf,gjfj2≥2λ-λ2Re⁡∑j∈J1-ωjf,gjfj,f+1-λ2Re⁡∑j∈Jωjf,gjfj,f.Proof.
We define the operatorsFω and F1-ω by (32)Fωf=∑j∈Jωjf,gjfj,F1-ωf=∑j∈J1-ωjf,gjfj. Then both series converge unconditionally and Fω,F1-ω∈B(H). Since Fω+F1-ω=IdH, by Corollary 2 we obtain (33)F1-ω∗F1-ωf,f+λFωf,f¯+λFωf,f≥2λ-λ2f2 for each f∈H. Hence (34)F1-ωf2+2λRe⁡Fωf,f≥2λ-λ2f,f. Therefore, (35)F1-ωf2≥2λ-λ2f,f-2λRe⁡Fωf,f=2λ-λ2Re⁡Fω+F1-ωf,f-2λRe⁡Fωf,f=2λ-λ2Re⁡F1-ωf,f-λ2Re⁡Fωf,f. It follows that (36)Re⁡∑j∈Jωjf,gjfj,f+∑j∈J1-ωjf,gjfj2=Re⁡Fωf,f+F1-ωf2≥2λ-λ2Re⁡F1-ωf,f+1-λ2Re⁡Fωf,f=2λ-λ2Re⁡∑j∈J1-ωjf,gjfj,f+1-λ2Re⁡∑j∈Jωjf,gjfj,f. This completes the proof.Remark 9.
If we takeλ=1/2 in Theorem 8, then we can obtain Theorem 3.3 in [14].



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*Source: 2108580-2018-10-03.xml*