On some equalities and inequalities for K frames

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ON SOME EQUALITIES AND INEQUALITIES FORK-FRAMES

Fahimeh Arabyani Neyshaburi∗, Ghadir Mohajeri Minaei∗∗and Ehsan Anjidani∗∗∗

∗_{Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran}

∗∗_{Technical and Vocational University, Neyshabur Branch, Iran}

∗∗∗_{Department of Mathematics, University of Neyshabur, P.O. Box 91136-899, Neyshabur, Iran}

e-mails: [email protected]; [email protected]; [email protected]

(Received 4 June 2017; 2016; after final revision 20 April 2018;

accepted 24 May 2018)

K-frame theory was recently introduced to reconstruct elements from the range of a bounded linear operator K in a separable Hilbert space. This significant property is worthwhile

espe-cially in some problems arising in sampling theory. Some equalities and inequalities have been

established for ordinary frames and their duals. In this paper, we continue and extend these

re-sults to obtain several important equalities and inequalities forK-frames. Moreover, by applying

Jensen’s operator inequality we obtain some new inequalities forK-frames.

Key words : Frames;K-frames; ParsevalK-frames;K-duals, Jensen’s operator inequality.

1. INTRODUCTION ANDPRELIMINARIES

Frames are redundant systems in separable Hilbert spaces, which provide non-unique representations

of vectors and are applied in a wide range of applications [4-7]. Working on efficient algorithms

for computing reconstruction of signals without noisy phase, Balan et al. [3] discover some new

identities for Parseval frames. Then, these identities have been generalized by some researchers for

the alternate dual frames and other types of frames [2, 11, 18, 19].

Recently, K-frames were introduced by G˘avrut¸a [13] to study atomic systems with respect to a

bounded operatorK ∈B(H). Recall that atomic decomposition for a closed subspaceH0of a Hilbert

spaceHintroduced by Feichtinger et al. [10] with frame-like properties. Although, the sequences

in atomic decompositions do not necessarily belong to H0. This interesting property, which was

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problems in sampling theory [15, 16]. In this work, we extend and improve these results to obtain

some equalities and inequalities forK-frames. First, we are going to state some preliminaries of

K-frames and their duals, which are used in our main results. LetHbe a separable Hilbert space and

I a countable index set andK ∈B(H). A sequenceF := {fi}i∈I ⊆ His called aK-frame forH,

if there exist constantsA, B >0such that

AkK∗fk2 ≤X

i∈I

|hf, fii|2≤Bkfk2, (f ∈ H). (1.1)

Obviously, F is an ordinary frame if K = IH and so K-frames are a generalization of the

ordinary frames [7, 8, 13]. The constantsAandBin (1.1) are called the lower and the upper bounds

ofF, respectively. AK-frameF is called a ParsevalK-frame, wheneverP_i∈I|hf, fii|2 =kK∗fk2.

Since F is a Bessel sequence, similar to ordinary frames the synthesis operator can be defined as

TF : l2 → H; TF({ci}i∈I) = P

i∈Icifi. The operator TF is bounded and its adjoint which is called the analysis operator is given byT_F∗(f) = {hf, fii}i∈I, and the frame operator is given by

S_F : H → H; SFf = TFTF∗f = P

i∈Ihf, fiifi. Unlike ordinary frames, the frame operator of aK-frame is not invertible in general. However, ifK has closed range, thenSF fromR(K) onto

S_F(R(K))is an invertible operator [17].

In [1], the notion of duality forK-frames is introduced and several approaches for construction

and characterization ofK-frames and their dual are presented. Indeed, a Bessel sequence{gi}i∈I ⊆

His called aK-dual of{fi}i∈Iif

Kf =X

i∈I

hf, giifi, (f ∈ H). (1.2)

Theorem 1.1 — (Douglas [9]). LetL1 ∈ B(H1,H) andL2 ∈ B(H2,H) be bounded linear

mappings on given Hilbert spaces. Then the following assertions are equivalent:

(i) R(L1)⊆R(L2);

(ii) L1L∗1≤λ2L2L∗2, for someλ >0;

(iii) There exists a bounded linear mappingX∈B(H1,H2), such thatL1 =L2X.

Moreover, if (i), (ii) and (iii) are valid, then there exists a unique operatorXso thatL1 =L2X

and

(a) kXk2 = inf{α >0, L1L∗1 ≤αL2L∗2};

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(c) R(X)⊂R(L∗₂).

Remark 1.2 : Suppose F = {fi}i∈I is a K-frame for H with the optimal bounds A and B,

respectively. ClearlyB=kSFk. Also, the lower inequality (1.1) implies that

AKK∗≤SF =TFTF∗.

Hence, by taking L1 = K andL2 = TF in Douglas’ theorem, it follows that there exists an

operatorX ∈B(H, l2)such that

TFX=K, (1.3)

and{X∗δi}i∈I is a K-dual ofF, where {δi}i∈I is the standard orthonormal basis of l2, see [13].

Moreover, by Douglas’ theorem the equation (1.3) has a unique solution asXF such that

kXFk2 = inf{α >0, KK∗ ≤αTFTF∗}. (1.4)

Now, letAbe the optimal lower bound ofF. Then, we obtain

A = sup{α >0 : αKK∗f ≤T_FT_F∗}

= (inf{β >0 : KK∗ ≤βTFTF∗})−1

= kXFk−2,

which coincides with ordinary frames. Indeed, ifK = IH, we easily obtainXF = T_F∗S_F−1 and so

kX_Fk−2 ₌_k_T∗

FSFk−2 =kSF−1k−1.

For further information inK-frame theory we refer the reader to [1, 10, 13, 17]. Throughout this

paper, we supposeHis a separable Hilbert space,K†the pseudo inverse of operatorK,Ia countable

index set and for everyJ ⊂ I, we denote the complement ofJ byJc. AlsoIHdenotes the identity

operator onH. For two Hilbert spacesH1 andH2 we denote byB(H1,H2)the set of all bounded

linear operators between H1 and H2, and we abbreviate B(H,H) by B(H). Also we denote the

range ofK∈B(H)byR(K), and the orthogonal projection ofHonto a closed subspaceV ofHby

π_V. Finally, we use ofTJ andSJ to denote the synthesis operator and frame operator, whenever the

index set is limited toJ ⊂I.

2. MAINRESULTS

In this section we obtain several equalities and inequalities forK-frames,K-duals and ParsevalK

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Theorem 2.1 — LetF = {fi}i∈I be aK-frame forHand{gi}i∈I be aK-dual ofF. Then for

everyJ ⊆I andf ∈ H, Ã

X

i∈J

hf, giihKf, fii ! − ° ° ° ° ° X i∈J

hf, giifi ° ° ° ° ° 2 = Ã X i∈Jc

hf, giihKf, fii ! − ° ° ° ° ° X i∈Jc

hf, giifi ° ° ° ° ° 2 .

PROOF: Suppose thatJ ⊆I, and consider the operator

MJf = X

i∈J

hf, giifi, (f ∈ H).

One can easily show thatMJ is a well defined and bounded operator onH. Moreover, we have

MJ +MJc =K. Hence, Ã

X

i∈J

hf, giihKf, fii ! − ° ° ° ° ° X i∈J

hf, giifi ° ° ° ° ° 2

= hK∗M_Jf, fi − hM_J∗M_Jf, fi

= h(K∗−M_J∗)MJf, fi

= hM_J∗c(K−MJc)f, fi

= hM_J∗cKf, fi − hM_J∗cM_Jcf, fi

= hf, K∗MJcfi − kM_Jcfk2

= Ã

X

i∈Jc

hf, giihKf, fii ! − ° ° ° ° ° X i∈Jc

hf, giifi ° ° ° ° ° 2 .2

Using Theorem 2.1 we immediately obtain the following corollary.

Corollary 2.2 — LetF ={fi}i∈I be aK-frame with aK-dual{gi}i∈I. Then for everyJ ⊆I

andf ∈ H,

hT_J{(X_Ff)_i}_i∈J, Kfi − kT_J{(X_Ff)_i}_i∈Jk2

= hT_Jc{(X_Ff)_i}_i∈Jc, Kfi − kT_Jc{(X_Ff)_i}_i∈Jck2,

whereXF is as in (1.4).

Theorem 2.3 — LetF = {fi}i∈I be aK-frame forHand{gi}i∈I be aK-dual ofF. Then for

every bounded sequence{αi}i∈Iandf ∈ H, Ã

X

i∈I

αihf, giihKf, fii ! − ° ° ° ° ° X i∈I

αihf, giifi ° ° ° ° ° 2 = Ã X i∈I

(1−α_i)hf, g_iihKf, f_ii ! − ° ° ° ° ° X i∈I

(1−α_i)hf, g_iif_i

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In the sequel, we survey some inequalities on ParsevalK-frames. These results also extend some

important inequalities for frames in [2, 11].

Theorem 2.4 — Assume thatF ={fi}i∈Iis a ParsevalK-frame forH. For everyf ∈ H,J ⊆I

andE ⊆Jcwe have

° ° ° ° ° X i∈J∪E

hf, f_iif_i

° ° ° ° ° 2 − ° ° ° ° ° ° X

i∈Jc_\E

hf, f_iif_i

° ° ° ° ° ° 2 = ° ° ° ° ° X i∈J

hf, f_iif_i

° ° ° ° ° 2 − ° ° ° ° ° X i∈Jc

hf, f_iif_i

° ° ° ° ° 2

+ 2ReX

i∈E

hf, f_iihKK∗_{f, f} ii.

PROOF: For the subsetJ ofI, we denote bySJ the operator defined bySJf = P

i∈Jhf, fiifi. Hence we haveSJ +SJc =KK∗. HenceS_J2−S_J2c =KK∗S_J −S_JcKK∗. In fact

S_J2−S2_Jc = S_J2−(KK∗−S_J)2

= KK∗SJ+SJKK∗−(KK∗)2

= KK∗SJ−(KK∗−SJ)KK∗

= KK∗SJ−SJcKK∗.

Hence, for everyf ∈ Hwe obtain

kS_Jfk2− kS_Jcfk2 =hKK∗S_Jf, fi − hS_JcKK∗f, fi,

and consequently ° ° ° ° ° X i∈J∪E

hf, fiifi ° ° ° ° ° 2 − ° ° ° ° ° ° X

i∈Jc_\E

hf, fiifi ° ° ° ° ° ° 2 = X i∈J∪E

hf, fiihKK∗f, fii − X

i∈Jc_\E

hf, fiihKK∗f, fii

= X

i∈J

hf, f_iihKK∗_{f, f} ii −

X

i∈Jc

hf, f_iihKK∗_{f, f}

ii+ 2Re X

i∈E

hf, f_iihKK∗_{f, f} ii = ° ° ° ° ° X i∈J

hf, f_iif_i

° ° ° ° ° 2 − ° ° ° ° ° X i∈Jc

hf, f_iif_i

° ° ° ° ° 2

+ 2ReX

i∈E

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Theorem 2.5 — Let F = {fi}i∈I be a ParsevalK-frame for H. Then for everyJ ⊆ I and

f ∈ H,

Re

Ã X

i∈Jc

hf, fiihKK∗f, fii ! + ° ° ° ° ° X i∈J

hf, fiifi ° ° ° ° ° 2 = Re Ã X i∈J

hf, fiihKK∗f, fii ! + ° ° ° ° ° X i∈Jc

hf, fiifi ° ° ° ° ° 2 ≥ 3

4kKK ∗_f_k2_.

PROOF: SinceS_J2−S2_Jc =KK∗SJ −SJcKK∗, we can write

S_J2+S_J2c = 2 µ

KK∗ 2 −SJ

¶₂

+ (KK∗)2

2 ≥

(KK∗₎2

2 ,

and consequently

KK∗SJ+SJ2c+ (KK∗S_J+S2_Jc)∗

= KK∗SJ+SJ2c+S_JKK∗+S_J2c

= KK∗(SJ+SJc) +S_J2+S_J2c

= (SJ+SJc)KK∗+S_J2+S_J2c ≥ 3 2(KK

∗₎2_.

Thus, we obtain

Re

Ã X

i∈Jc

hf, fiihKK∗f, fii ! + ° ° ° ° ° X i∈J

hf, fiifi ° ° ° ° ° 2 = Re Ã X i∈J

hf, f_iihKK∗_{f, f} ii ! + ° ° ° ° ° X i∈Jc

hf, f_iif_i

° ° ° ° ° 2 = 1 2 ¡

hKK∗S_Jf, fi+hS_J2cf, fi+hf, KK∗S_Jfi+hf, S2_Jcfi ¢

≥ 3

4kKK ∗_f_k2_.

This completes the proof. 2

Notice that, in Theorem 2.5 if we takeK =IHit reduce to Theorem 2.2 [19]. Also, Theorem 2.5

leads to the following concept, which is a generalization of [11] for Parseval frames. LetF ={fi}i∈I

be a ParsevalK-frame. Then we consider

v₊(F, K, J) = sup f6=0

Re³P_i∈Jchf, f_iihKK∗f, f_ii ´

+°°P_i∈Jhf, f_iif_i°°2

kKK∗_f_k2 ,

v₋(F, K, J) = inf f6=0

Re³P_i∈Jchf, f_iihKK∗f, f_ii ´

+°°P_i∈Jhf, f_iif_i°°2

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Now, we are going to present some properties of these notations. For this we need the next lemma.

Lemma 2.6 — LetK be a closed range operator andF ={fi}i∈Ibe aK-frame forH. Then

(i) °°P_i∈Ihf, fiifi ° °2 _{≤ k}_S

Fk P

i∈I|hf, fii|2, (f ∈ H);

(ii) P_i∈I|hf, fii|2 ≤ kK†k2kXFk2 ° °P

i∈Ihf, fiifi °

°2_, ₍_f _∈ _R₍_K_{)), where}_X

F is as in equa-tion (1.1).

PROOF : The first part is a known result for every Bessel sequence [8] and so for K-frames.

Hence, we only need to show(ii). Letf ∈R(K). Then we have Ã

X

i∈I

|hf, fii|2 !₂

= |hSFf, fi|2

≤ kSFfk2kfk2

= kSFfk2k(K†)∗K∗fk2

≤ kSFfk2k(K†)k2kK∗fk2

≤ kS_Ffk2k(K†)k2kX_Fk2X

i∈I

|hf, f_ii|2.2

Theorem 2.7 — SupposeF = {fi}i∈I is a ParsevalK-frame forH. The following assertion

hold:

(i) 3

4 ≤v−(F, K, J)≤v+(F, K, J)≤ kKkkK

†_k_{(1 +}_k_K_k₎

(ii) v+(F, K, J) =v+(F, K, Jc), andv−(F, K, J) =v−(F, K, Jc)

(iii) v+(F, K, I) =v−(F, K, I) = 1, andv+(F, K,∅) =v−(F, K,∅) = 1.

PROOF: For (i), it is sufficient to show the upper inequality. SinceF is a Bessel sequence, so by

applying Lemma 2.6 (i) we obtain ° ° ° ° °

X

i∈J

hf, fiifi ° ° ° ° °

2

≤ kSJk X

i∈J

|hf, fii|2

≤ kS_JkX

i∈I

|hf, fii|2

≤ kKk2kK∗fk2

= kKk2kK†KK∗fk2

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Moreover,

Re

Ã X

i∈Jc

hf, fiihKK∗f, fii !

≤ Ã

X

i∈I

|hf, fii|2 !₁_/₂Ã

X

i∈I

|hKK∗f, fii|2 !₁_/₂

= kK∗fkkK∗KK∗fk

= kK†KK∗fkkK∗KK∗fk

≤ kKkkK†kkKK∗fk2.

Hence,

v−(F, K, J)≤v+(F, K, J)≤ kKkkK†k(1 +kKkkK†k).

On the other hand, in the proof of Theorem 2.4 we observed that

KK∗SJ−SJcKK∗ =S_J2 −S_J2c.

and consequently we have

hS_J2f, fi+hSJcKK∗f, fi=hS_J2cf, fi+hKK∗S_Jf, fi,

for everyf ∈ H. Thus

° ° ° ° °

X

i∈J

hf, fiifi ° ° ° ° °

2

+X

i∈Jc

= ° ° ° ° °

X

i∈Jc

hf, fiifi ° ° ° ° °

2

+X

i∈J

hf, fiihKK∗f, fii.

This shows (ii). Finally (iii) is easy to check. 2

Using the result above-mentioned, we establish some equivalent results for ParsevalK-frames.

Corollary 2.8 — Let{fi}i∈I be a ParsevalK-frame forH. Then for everyJ ⊂I andf ∈ Hthe

following conditions are equivalent.

(i) v+(F, K, J) =v−(F, K, J) = 1.

(ii) °°P_i∈Jhf, fiifi °

°2₌_ReP

i∈Jhf, fiihKK∗f, fii.

(iii) °°P_i∈Jchf, f_iif_i ° °2

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PROOF: (ii) ⇔ (iii) is clear. Also,(i) ⇒ (ii) holds by a direct computation. Now, let(ii)

holds, then

X

i∈Jc

hf, fiihKK∗f, fii+ ° ° ° ° °

X

i∈J

hf, fiifi ° ° ° ° °

2

= X

i∈I

= hKK∗f, SFfi

= kKK∗fk2.

i.e.,(i)holds. Hence(i)⇔(iii)and similarly(i)⇔(iii). 2

Finally, we obtain the following more general result.

Corollary 2.9 — Let{fi}i∈I be a ParsevalK-frame forH. Then for everyJ ⊂I andf ∈ Hthe

following conditions are equivalent.

(i) °°P_i∈Jhf, fiifi ° °2

=P_i∈Jhf, fiihKK∗f, fii.

(ii) °°P_i∈Jchf, fiifi ° °2

=P_i∈Jchf, fiihKK∗f, fii.

(iii) SJf ⊥SJcf_.

(iv) f ⊥SJcS_Jf.

PROOF: Since

kSJfk2− hKK∗SJf, fi=kSJcfk2− hKK∗S_Jcf, fi,

we have(i)⇔(ii). Moreover,SJ andSJcare positive, so we have

hS_Jcf, S_Jfi=hf, S_JcS_Jfi=h(S_JKK∗−S_J2)f, fi, (f ∈ H).

This follows that,(iii)⇔(iv)and(i)⇔(iv). 2

3. SOMENEWINEQUALITIES

In this section, applying Jensen’s operator inequality we obtain some new inequalities forK-frames.

First, recall that a continuous functionhdefined on an intervalLis said to be operator convex if

h(λA+ (1−λ)B)≤λh(A) + (1−λ)h(B),

for all λ ∈ [0,1] and all self-adjoint operatorsA,B with spectra in L. A general formulation of

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Theorem 3.1 — LetH,Kbe Hilbert spaces andh:L→Rbe an operator convex function. For

eachn∈N, the inequality

h( n X

i=1

Φi(Ai))≤ n X

i=1

Φi(h(Ai)), (3.1)

holds for everyn-tuple(A1, ..., An)of self-adjoint operators inB(H)with spectra inL and every

n-tuple(Φ1, ...,Φn)of positive linear mappingsΦi :B(H)→B(K)such that P_n

i=1Φi(IH) =IK.

Also, a variant of Jensen’s operator inequality (3.1) for convex functions is presented as follows

(see [14]).

Theorem 3.2 — LetA1, ..., Anbe self-adjoin operators with spectra in[m, M]for some scalars

m < MandΦ1, ...,Φnbe positive linear mappings fromB(H)intoB(K)with P_n

i=1Φi(IH) =IK, where IH andIK are the identity mappings on H andK, respectively. If h : [m, M] → R is a

continuous convex function, then

h

Ã

mI_K+M I_K−

n X

i=1

Φi(Ai) !

≤h(m)I_K+h(M)I_K−

n X

i=1

Φi(h(Ai)). (3.2)

Now, we apply inequalities (3.1) and (3.2) to obtain some inequalities forK-frames.

Theorem 3.3 — SupposeF ={fi}i∈I is a ParsevalK-frame forH. Then for every subsetJ of

I,

(i) ifh: [0,kKk2]→Ris an operator convex function, then

h

µ

KK∗ 2

¶

≤ h(SJ) +h(SJc)

2 ; (3.3)

(ii) ifh: [0,kKk2]→Ris a convex function, then

h

µ

kKk2I_H−KK∗

2 ¶

≤h(kKk2I_H)−h(SJ) +h(SJc)

2 . (3.4)

PROOF: LetA1=SJ andA2 =SJc. SinceS_J andS_Jc are positive operators withS_J+S_Jc =

KK∗_{, the spectra of}_A

1 andA2 are in[0,kKk2]. Now, define Φ1 = 1₂IB(H) andΦ2 = 1₂IB(H),

whereIB(H)is the identity mapping onB(H), and putm= 0andM =kKk2. Then, assertions(i)

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Corollary 3.4 — SupposeF ={fi}i∈I is a ParsevalK-frame forH. Then for every subsetJ of

Iandf ∈ H, we obtain

1 2kKK

∗_f_k2 _≤

° ° ° ° °

X

i∈J

hf, f_iif_i

° ° ° ° °

2

+ ° ° ° ° °

X

i∈Jc

hf, f_iif_i

° ° ° ° °

2

≤ 2kKk2kK∗fk2−1

2kKK

∗_f_k2_. _(3.5)

PROOF: Since the functionh(x) =x2is convex on[0,∞), by Theorem 3.3 we have

(KK∗₎2

4 ≤

S_J2+S_J2c

2 , (3.6)

and _µ

kKk2IH−KK ∗

2 ¶₂

≤ kKk4IH− S

2

J+SJ2c

2 ,

and so

S2

J +SJ2c

2 ≤ kKk

2_KK∗₋(KK∗)2

4 . (3.7)

Therefore, it follows from (3.6) and (3.7) that

(KK∗₎2

2 ≤S

2

J +SJ2c ≤2kKk2KK∗−

(KK∗₎2

2 .

Hence for everyf ∈ H, we obtain inequality (3.5). 2

ACKNOWLEDGEMENT

The authors would like to thank the referee for helpful comments which improved the paper.

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