Laura G˘AVRUT¸APa¸sc G˘AVRUT¸A

Politehnica University of Timi¸soara，Department of Mathematics，Victoriei Square no.2，300006 Timi¸soara，Romania

E-mail∶gavruta laura@yahoo.com；pgavruta@yahoo.com



SOME PROPERTIES OF OPERATOR-VALUED FRAMES∗

Laura G˘AVRUT¸A†Pa¸sc G˘AVRUT¸A

Politehnica University of Timi¸soara，Department of Mathematics，Victoriei Square no.2，300006 Timi¸soara，Romania

E-mail∶gavruta laura@yahoo.com；pgavruta@yahoo.com

Operator-valued frames（or g-frames）are generalizations of frames and fusion frames and have been used in packets encoding，quantum computing，theory of coherent states and more.In this article，we give a new formula for operator-valued frames for finite dimensional Hilbert spaces.As an application，we derive in a simple manner a recent result of A.Najati concerning the approximation of g-frames by Parseval ones.We obtain also some results concerning the best approximation of operator-valued frames by its alternate duals，with optimal estimates.

Frames；g-frames

2010 MR Subject Classification42C15

1　Introduction

Frames in Hilbert spaces were introduced by Duffin and Schaeffer［10］in 1952，in the context of nonharmonic Fourier series.After a couple of years，in 1986，frames were brought to life by Daubechies，Grossman，and Meyer［9］.Frames have nice properties which makes them useful tools in signal processing，image processing，coding theory，sampling theory and more.

In the following，we denote by H a separable Hilbert space and by L（H）the space of all linear bounded operators on H.

Definition 1.1A family of elements⊂H is called a frame for H if there exist constants A，B＞0 such that

The constants A，B are called frame bounds.

We say that a frame is tight if A=B，a Parseval frame if A=B=1，and an exact frame if it ceases to be a frame when any one of its elements is removed.

The exact frames are in fact Riesz bases.If just the last inequality in the above definition holds，we say thatis a Bessel sequence.

is called synthesis operator（or pre-frame operator）.Its adjoint operator is given by

and is called the analysis operator.By composing T with its adjoint T∗，we obtain the frame operator

The next theorem is one of the most important results about frames.

（i）S is invertible and self-adjoint；

（ii）every x∈H can be represented as

The relation（1.1）is called the reconstruction formula.We call｛〈x，the frame coefficients.

A generalization of frames，which allows to reconstruct elements from the range of a linear and bounded operator in a Hilbert space，was obtain by L.G˘avrut¸a［13］.

In 2006，W.Sun［21］introduced the concept of g-frame.g-frames are generalized frames，which include ordinary frames，bounded invertible linear operators，fusion frames，as well as many recent generalizations of frames.See also the article of V.Kaftal，D.Larson and S.Zhang［18］.For the general theory of fusion frames，see the article of P.G.Casazza et al［6］and P. G˘avrut¸a［16］.

For the connection between the theory of g-frames and quantum theory as in［7，19］，see the papers［1］and［17］.

In the following，we consider H and K to be two Hilbert spaces.We denote by L（H，K）the space of all linear bounded operators from H into K.By I we denote a finite or a countable set.

Definition 1.3We say that a sequence｛Λi∈L（H，K）：i∈I｝is a generalized frame or a g-frame for H if there exist two positive constants A and B such that

We call A and B frame bounds.We say that｛Λi：i∈I｝is a g-tight frame if A=B and a g-Parseval frame if A=B=1.

A frame is equivalent to a g-frame whenever K=C.

The g-frame operator S is defined as follows whereis the adjoint operator of Λi.

W.Sun proved in［21］that S is well-defined，bounded，and self-adjoint operator.Then，the following reconstruction formula takes place for all x∈H

We call｛ΛiS−1｝the canonical dual g-frame of｛Λi｝.A g-frame｛Γi｝is called an alternate dual g-frame of｛Λi｝if it satisfies

2　Results

Before the main results，we give some preliminary results（we refer to Propositions 2.1，2.2，2.3，2.4）.

Proposition 2.1Let｛Λi｝∈L（H，K）be a g-frame.Then，H is finite-dimensional if and only if

where by‖·‖Fwe denote the Frobenius norm（or the Hilbert-Schmidt norm）.

If dim H=n=card J，we have，we have

Proposition 2.2If｛Λi｝and｛Γi｝are Parseval g-frames，then

ProofLet L∈L（H，K），then

We use the fact that‖T‖F=‖T∗‖F.

Proposition 2.3Let｛Γi｝be a Parseval g-frame and H a finite n-dimensional Hilbert space.Then，X

More general，we have the following result.

Proposition 2.4Let｛Λi｝be a g-frame for a finite n-dimensional Hilbert space H with g-frame operator S.Then，

The identity given in the next theorem was obtain for the particular case of ordinary frames in［14］and for the case of continuous frames in［15］.

Theorem 2.5Let｛Λi｝be a g-frame with g-frame operator S，｛Γi｝be a Parseval g-frame，and H be a finite-dimensional Hilbert space.Then，we have the following estimation：

ProofWe have

In contrast，

where we use Proposition 2.3.

As a corollary，we have immediately the following result of A.Najati［20］.

Corollary 2.6Let｛Λi｝be a g-frame，with g-frame operator S and H be a finitedimensional Hilbert space.Then，for all Parseval g-frames｛Γi｝，the following inequality，

holds，and the equality occurs if and only if，（∀）i∈I.

In the following，we consider 0≤ε＜1.The next definition extends the definition for ordinary frames of P.G.Casazza［3］.

Definition 2.7We say that｛Λi｝is ε nearly Parseval g-frame if for all x∈H，

Theorem 2.8If｛Λi｝is an ε nearly Parseval g-frame，with g-frame operator S and H is a finite n-dimensional Hilbert space，then Moreover，the estimation is optimal.

ProofLet S have eigenvalues｛λ1，λ2，···，λn｝and the corresponding orthonormal eigenvectors｛e1，e2，···，en｝.It follows that

Using（2.1），for x=ek，we obtain

and so 1−ε≤λk≤1+ε，which implies

We prove now that the estimation is optimal.Indeed，ifis an orthonormal basis of H and v∈K，a fixed vector with‖v‖=1，we take the operators Λk：H→C，Λkx=We have

In the following，we give analogous results for the case when a canonical Parseval g-frame is replaced by canonical dual frame.We have the following identity.

Proposition 2.9Let｛Λi｝be a g-frame，with g-frame operator S and｛Γi｝be an alternate dual g-frame of｛Λi｝.Then，， hence we have equality.

ProofWe have，for all x∈H，

because

Similarly，we have Using the fact that S is g-frame operator，that is，which implies〈Sx，x〉=By replacing x with S−1x，we get

Then by（2.4）and（2.5），we get

We also have

So，

If we add relations（2.6），（2.7）and use（2.3），we obtain

So，we obtained（2.2）.

Corollary 2.10Let｛Λi｝be a g-frame for the Hilbert space H，with frame operator S. For all alternate dual g-frames｛Γi｝of｛Λi｝，the inequality

takes place for any x∈H and the equality holds for any x∈H if and only if Γi=ΛiS−1，（∀）i∈I.

The following result is an analog of Theorem 2.5 for alternate duals.

Theorem 2.11Let｛Λi｝be a g-frame for H finite dimensional Hilbert space，with frame operator S and｛Γi｝to be an alternate dual of｛Λi｝.Then，

ProofWe have

From Proposition 2.9，we get

Corollary 2.12Let｛Λi｝be a g-frame for H finite dimensional Hilbert space，with frame operator S.Then，for every alternate dual g-frames｛Γi｝of｛Λi｝，the inequality

takes place，and the equality holds if and only if Γi=ΛiS−1，（∀）i∈I.

Theorem 2.13If｛Λi｝is a nearly Parseval g-frame and H is a finite n-dimensional Hilbert space，then there exists a dual｛Γi｝of｛Λi｝such that

Moreover，the estimation is optimal.

We have equality for the same g-frame as in example we constructed in the end of the proof of Theorem 2.8.

RemarksSome results of this article were presented at the 24thInternational Conference on Operator Theory，July 2-7，2012，West University of Timi¸soara，Romania.

Some results of this article are related to the ones presented in［11，12］.［11，12］deals only with vector frames.Our results are more general and we use a technique which is more simple even in the case of vector frames.In addition，our article contains results concerning the best approximation of operator-valued frames by its alternate duals，with optimal estimates.

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September 10，2014；revised June 6，2015.The final work of P.G˘avrut¸a on this article was supported by a grant of Romanian National Authority for Scientific Research，CNCS-UEFISCDI，project number PN-II-ID-JRP-2011-2/11-RO-FR/01.03.2013.

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