continuous -k-g-frame in hilbert -modulesdownloads.hindawi.com/journals/jfs/2019/2426978.pdf ·...

Research ArticleContinuous lowast-K-G-Frame in Hilbert 119862lowast-Modules

Mohamed Rossafi 1 Abdeslam Touri1 Hatim Labrigui1 and Abdellatif Akhlidj2

1Department of Mathematics University of Ibn Tofail B P 133 Kenitra Morocco2Department of Mathematics University of Hassan II Casablanca Morocco

Correspondence should be addressed to Mohamed Rossafi rossafimohamedgmailcom

Received 11 May 2019 Accepted 21 July 2019 Published 31 July 2019

Academic Editor Gestur Olafsson

Copyright copy 2019 Mohamed Rossafi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Frame theory is exciting and dynamic with applications to a wide variety of areas in mathematics and engineering In this paperwe introduce the concept of Continuous lowast-K-g-frame in Hilbert Clowast-Modules and we give some properties

1 Introduction and Preliminaries

The concept of frames in Hilbert spaces has been introducedby Duffin and Schaeffer [1] in 1952 to study some deep prob-lems in nonharmonic Fourier series after the fundamentalpaper [2] by Daubechies Grossman andMeyer frame theorybegan to be widely used particularly in the more specializedcontext of wavelet frames and Gabor frames [3]

Traditionally frames have been used in signal processingimage processing data compression and sampling theory Adiscreet frame is a countable family of elements in a separableHilbert space which allows for a stable not necessarilyunique decomposition of an arbitrary element into an expan-sion of the frame elements The concept of a generalizationof frames to a family indexed by some locally compact spaceendowed with a Radon measure was proposed by G Kaiser[4] and independently by Ali Antoine and Gazeau [5]Theseframes are known as continuous frames Gabardo and Hanin [6] called these frames associated with measurable spacesAskari-Hemmat Dehghan and Radjabalipour in [7] calledthem generalized frames and in mathematical physics theyare referred to as coherent states [5]

In this paper we introduce the notion of Continuouslowast-K-g-Frame which are generalization of lowast-K-g-Frame in Hilbert119862lowast-Modules introduced by M Rossafi and S Kabbaj [8] andwe establish some new results

The paper is organized as follows we continue thisintroductory sectionwe briefly recall the definitions and basicproperties of 119862lowast-algebra Hilbert 119862lowast-modules In Section 2we introduce the Continuous lowast-K-g-Frame the Continuous

pre-lowast-K-g-frame operator and the Continuous lowast-K-g-frameoperator also we establish here properties

In the following we briefly recall the definitions and basicproperties of 119862lowast-algebra HilbertA-modules Our referencefor 119862lowast-algebras is [9 10] For a 119862lowast-algebra A if 119886 isin A ispositive we write 119886 ge 0 and A+ denotes the set of positiveelements ofA

Definition 1 (see [11]) LetA be a unital 119862lowast-algebra andH aleftA-module such that the linear structures ofA andH arecompatible H is a pre-Hilbert A-module if H is equippedwith anA-valued inner product ⟨ ⟩A HtimesH 997888rarr A suchthat is sesquilinear positive definite and respects the moduleaction In the other words

(i) ⟨119909 119909⟩A ge 0 for all 119909 isinH and ⟨119909 119909⟩A = 0 if and onlyif 119909 = 0

(ii) ⟨119886119909 + 119910 119911⟩A = 119886⟨119909 119910⟩A + ⟨119910 119911⟩A for all 119886 isin A and119909 119910 119911 isinH

(iii) ⟨119909 119910⟩A = ⟨119910 119909⟩lowastA for all 119909 119910 isinH

For 119909 isin H we define 119909 = ⟨119909 119909⟩A12 If H is completewith it is called a Hilbert A-module or a Hilbert 119862lowast-module over A For every 119886 in 119862lowast-algebra A we have |119886| =(119886lowast119886)12 and the A-valued norm on H is defined by |119909| =⟨119909 119909⟩12A for 119909 isinH

Let H and K be two Hilbert A-modules A map 119879 H 997888rarr K is said to be adjointable if there exists a map119879lowast K 997888rarrH such that ⟨119879119909 119910⟩A = ⟨119909 119879lowast119910⟩A for all 119909 isinHand 119910 isinK

HindawiJournal of Function SpacesVolume 2019 Article ID 2426978 5 pageshttpsdoiorg10115520192426978

2 Journal of Function Spaces

We reserve the notation 119864119899119889lowastA(HK) for the set of alladjointable operators from H to K and 119864119899119889lowastA(HH) isabbreviated to 119864119899119889lowastA(H)

The following lemmas will be used to prove our mainsresults

Lemma 2 (see [11]) Let H be Hilbert A-module If 119879 isin119864119899119889lowastA(H) then

⟨119879119909 119879119909⟩ le 1198792 ⟨119909 119909⟩ forall119909 isinH (1)

Lemma 3 (see [12]) Let H and K two Hilbert A-Modulesand 119879 isin 119864119899119889lowast(HK) en the following statements areequivalent

(i) 119879 is surjective(ii) 119879lowast is bounded below with respect to norm ie there is

119898 gt 0 such that 119879lowast119909 ge 119898119909 for all 119909 isinK(iii) 119879lowast is bounded below with respect to the inner product

ie there is 1198981015840 gt 0 such that ⟨119879lowast119909 119879lowast119909⟩ ge 1198981015840⟨119909 119909⟩for all 119909 isinK

Lemma 4 (see [13]) LetH andK be two HilbertA-Modulesand 119879 isin 119864119899119889lowast(HK) en

(i) if 119879 is injective and 119879 has closed range then theadjointable map 119879lowast119879 is invertible and

10038171003817100381710038171003817(119879lowast119879)minus110038171003817100381710038171003817minus1 119868H le 119879lowast119879 le 1198792 119868H (2)

(ii) If 119879 is surjective then the adjointable map 119879119879lowast isinvertible and

10038171003817100381710038171003817(119879119879lowast)minus110038171003817100381710038171003817minus1 119868K le 119879119879lowast le 1198792 119868K (3)

2 Continuous lowast-K-g-Frame in Hilbert119862lowast-Modules

Let119883be a Banach space (Ω 120583) ameasure space and function119891 Ω 997888rarr 119883 a measurable function Integral of the Banach-valued function 119891 has defined Bochner and others Mostproperties of this integral are similar to those of the integral ofreal-valued functions Because every 119862lowast-algebra and Hilbert119862lowast-module is a Banach space thus we can use this integraland its properties

Let (Ω 120583) be a measure space let119880 and 119881 be two Hilbert119862lowast-modules 119881119908 119908 isin Ω is a sequence of subspaces of Vand 119864119899119889lowastA(119880119881119908) is the collection of all adjointable A-linearmaps from119880 into 119881119908 We define

⨁119908isinΩ

119881119908

= 119909 = 119909119908 119909119908 isin 1198811199081003817100381710038171003817100381710038171003817intΩ

100381610038161003816100381611990911990810038161003816100381610038162 119889120583 (119908)1003817100381710038171003817100381710038171003817 lt infin

(4)

For any 119909 = 119909119908 119908 isin Ω and 119910 = 119910119908 119908 isinΩ if the A-valued inner product is defined by ⟨119909 119910⟩ =intΩ⟨119909119908 119910119908⟩119889120583(119908) the norm is defined by 119909 = ⟨119909 119909⟩12

the⨁119908isinΩ119881119908 is a Hilbert 119862lowast-module

Definition 5 Let119870 isin 119864119899119889lowastA(119880) we call Λ119908 isin 119864119899119889lowastA(119880119881119908) 119908 isin Ω a Continuous lowast-K-g-frame for Hilbert 119862lowast-module119880with respect to 119881119908 119908 isin Ω if

(a) for any 119909 isin 119880 the function 119909 Ω 997888rarr 119881119908 defined by119909(119908) = Λ119908119909 is measurable

(b) there exist two strictly nonzero elements 119860 and 119861 inA such that

119860⟨119870lowast119909119870lowast119909⟩119860lowast le intΩ⟨Λ119908119909 Λ119908119909⟩ 119889120583 (119908)

le 119861 ⟨119909 119909⟩ 119861lowast forall119909 isin 119880(5)

The elements 119860 and 119861 are called Continuous lowast-K-g-framebounds

If 119860 = 119861 we call this Continuous lowast-K-g-frame a con-tinuous tight lowast-K-g-frame and if 119860 = 119861 = 1A it is calleda continuous Parseval lowast-K-g-frame If only the right-handinequality of (5) is satisfied we call Λ119908 119908 isin Ω acontinuous lowast-K-g-Bessel for 119880 with respect to Λ119908 119908 isin Ωwith Bessel bound 119861Example 6 Let 119897infin be the set of all bounded complex-valuedsequences For any 119906 = 119906119895119895isinN V = V119895119895isinN isin 119897infin we define

119906V = 119906119895V119895119895isinN 119906lowast = 119906119895119895isinN 119906 = sup

119895isinN

1003816100381610038161003816100381611990611989510038161003816100381610038161003816

(6)

ThenA = 119897infin is a Clowast-algebraLetH = 1198620 be the set of all sequences converging to zero

For any 119906 V isinH we define

⟨119906 V⟩ = 119906Vlowast = 119906119895119906119895119895isinN (7)

ThenH is a HilbertA-moduleDefine 119891119895 = 119891119895119894 119894isinNlowast by 119891119895119894 = 12 + 1119894 if 119894 = 119895 and 119891119895119894 = 0

if 119894 = 119895 forall119895 isin NlowastNow define the adjointable operator Λ 119895 H 997888rarr

A Λ 119895119909 = ⟨119909 119891119895⟩Then for every 119909 isinH we have

sum119895isinN⟨Λ 119895119909 Λ 119895119909⟩ = 12 +

1119894 119894isinNlowast ⟨119909 119909⟩

12 +

1119894 119894isinNlowast (8)

So Λ 119895119895 is a 12 + 1119894119894isinNlowast-tight lowast-g-frameLet119870 H 997888rarrH defined by 119870119909 = 119909119894119894119894isinNlowast Then for every 119909 isinH we have

⟨119870lowast119909119870lowast119909⟩Ale sum119895isinN⟨Λ 119895119909 Λ 119895119909⟩

= 12 +1119894 119894isinNlowast ⟨119909 119909⟩

12 +

1119894 119894isinNlowast

(9)

Now let (Ω 120583) be a 120590-finite measure space with infinitemeasure and 119867120596120596isinΩ be a family ofHilbertA-module (119867120596 =1198620 forall119908 isin Ω)

Journal of Function Spaces 3

Since Ω is a 120590-finite it can be written as a disjoint unionΩ = ⋃Ω120596 of countably many subsets Ω120596 sube Ω such that120583(Ω119896) lt infin forall119896 isin N Without less of generality assume that120583(Ω119896) gt 0forall119896 isin N

For each 120596 isin Ω define the operator Λ 120596 119867 997888rarr 119867119908 by

Λ119908 (119909) = 1120583 (Ω119896) ⟨119909 119891119896⟩ ℎ120596 forall119909 isin 119867 (10)

where 119896 is such that 119908 isin Ω120596 and ℎ120596 is an arbitrary element of119867120596 such that ℎ120596 = 1

For each 119909 isin 119867 Λ 120596119909120596isinΩ is strongly measurable (sinceℎ120596 are fixed) and

intΩ⟨Λ 120596119909 Λ 120596119909⟩ 119889120583 (120596) = sum

119895isinN

⟨119909 119891119895⟩ ⟨119891119895 119909⟩ (11)

So therefore

⟨119870lowast119909119870lowast119909⟩ le intΩ⟨Λ 120596119909 Λ 120596119909⟩ 119889120583 (120596)

= sum119895isinN

⟨119909 119891119895⟩ ⟨119891119895 119909⟩

= 12 +1119894 119894isinNlowast ⟨119909 119909⟩

12 +


(12)

So Λ 120596120596isinΩ is a continuous lowast-K-g-frame

Remark 7

(i) Every continuous lowast-g-frame is a continuous lowast-K-g-frame indeedLet Λ119908 isin 119864119899119889lowastA(119880119881119908) 119908 isin Ω be a continuouslowast-g-frame for Hilbert 119862lowast-module 119880 with respect to119881119908 119908 isin Ω then

119860⟨119909 119909⟩ 119860lowast le intΩ⟨Λ119908119909 Λ119908119909⟩ 119889120583 (119908) le 119861 ⟨119909 119909⟩ 119861lowast

forall119909 isin 119880(13)

or

⟨119870lowast119909119870lowast119909⟩ le 1198702 ⟨119909 119909⟩ forall119909 isin 119880 (14)

then

(119870minus1 119860) ⟨119870lowast119909 119870lowast119909⟩ (119870minus1 119860)lowast

le intΩ⟨Λ119908119909 Λ119908119909⟩ 119889120583 (119908) le 119861 ⟨119909 119909⟩ 119861lowast

(15)

so let Λ119908 isin 119864119899119889lowastA(119880119881119908) 119908 isin Ω be a continuouslowast-K-g-frame with lower and upper bounds 119870minus1119860and 119861 respectively

(ii) If 119870 isin 119864119899119889lowastA(119867) is a surjective operator then everycontinuous lowast-K-g-frame for 119867 with respect to 119881119908 119908 isin Ω is a continuous lowast-g-frame

Indeedif 119870 is surjective there exists 119898 gt 0 such that

119898⟨119909 119909⟩ le ⟨119870lowast119909119870lowast119909⟩ (16)

then

(119860radic119898) ⟨119909 119909⟩ (119860radic119898)lowast le 119860⟨119870lowast119909119870lowast119909⟩119860lowast (17)

or if Λ119908 isin 119864119899119889lowastA(119880119881119908) 119908 isin Ω is a continuouslowast-K-g-frame we have

(119860radic119898) ⟨119909 119909⟩ (119860radic119898)lowast le intΩ⟨Λ119908119909 Λ119908119909⟩ 119889120583 (119908)

le 119861 ⟨119909 119909⟩ 119861lowast(18)

hence Λ119908 isin 119864119899119889lowastA(119880119881119908) 119908 isin Ω is a continuouslowast-g-frame for 119880 with lower and upper bounds 119860radic119898and 119861 respectively

Let 119870 isin 119864119899119889lowastA(119880) and Λ119908 isin 119864119899119889lowastA(119880119881119908) 119908 isin Ωbe a continuous lowast-K-g-frame for Hilbert 119862lowast-module 119880 withrespect to 119881119908 119908 isin Ω

We define an operator 119879 119880 997888rarr⨁119908isinΩ119881119908 by119879119909 = Λ119908119909 119908 isin Ω forall119909 isin 119880 (19)

then 119879 is called the continuous lowast-K-g-frame transformSo its adjoint operator is 119879lowast ⨁119908isinΩ119881119908 997888rarr 119880 given by

119879lowast (119909120596120596isinΩ) = intΩΛlowast120596119909120596119889120583 (119908) (20)

By composing 119879 and 119879lowast the frame operator 119878 = 119879lowast119879 givenby

119878119909 = intΩΛlowast120596Λ 120596119909119889120583(119908) S is called continuous lowast-K-g

frame operator

Theorem 8 e continuous lowast-K-g frame operator 119878 is bound-ed positive self-adjoint and 119860minus1minus21198702 le 119878 le 1198612

Proof First we show 119878 is a self-adjoint operator By definitionwe have forall119909 119910 isin 119880

⟨119878119909 119910⟩ = ⟨intΩΛlowast119908Λ119908119909119889120583 (119908) 119910⟩

= intΩ⟨Λlowast119908Λ119908119909 119910⟩ 119889120583 (119908)

= intΩ⟨119909 Λlowast119908Λ119908119910⟩ 119889120583 (119908)

= ⟨119909 intΩΛlowast119908Λ119908119910119889120583 (119908)⟩ = ⟨119909 119878119910⟩

(21)

Then 119878 is a self-adjointClearly 119878 is positiveBy definition of a continuous lowast-K-g-frame we have


le 119861 ⟨119909 119909⟩ 119861lowast(22)


So

119860⟨119870lowast119909 119870lowast119909⟩119860lowast le ⟨119878119909 119909⟩ le 119861 ⟨119909 119909⟩ 119861lowast (23)

This gives10038171003817100381710038171003817119860minus1

10038171003817100381710038171003817minus2 1003817100381710038171003817⟨119870119870lowast119909 119909⟩1003817100381710038171003817 le ⟨119878119909 119909⟩ le 1198612 ⟨119909 119909⟩ (24)

If we take supremum on all 119909 isin 119880 where 119909 le 1 wehave

10038171003817100381710038171003817119860minus110038171003817100381710038171003817minus2 1198702 le 119878 le 1198612 (25)

Theorem 9 Let 119870 isin 119864119899119889lowastA(119867) be surjective and Λ119908 isin119864119899119889lowastA(119880119881119908) 119908 isin Ω a continuous lowast-K-g-frame for 119880 withlower and upper bounds 119860 and 119861 respectively and with thecontinuous lowast-K-g-frame operator 119878

Let 119879 isin 119864119899119889lowastA(119880) be invertible then Λ119908119879 119908 isin Ω isa continuous lowast-K-g-frame for 119880 with continuous lowast-K-g-frameoperator 119879lowast119878119879Proof We have



Using Lemma 3 we have (119879lowast119879)minus1minus1⟨119909 119909⟩ le ⟨119879119909 119879119909⟩ forall119909 isin119880

119870 is surjective then there exists119898 such that

119898⟨119879119909 119879119909⟩ le ⟨119870lowast119879119909119870lowast119879119909⟩ (27)

and then

11989810038171003817100381710038171003817(119879lowast119879)minus110038171003817100381710038171003817minus1 ⟨119909 119909⟩ le ⟨119870lowast119879119909119870lowast119879119909⟩ (28)

so

11989810038171003817100381710038171003817(119879lowast119879)minus110038171003817100381710038171003817minus1 119860⟨119909 119909⟩ 119860lowast le 119860⟨119870lowast119879119909119870lowast119879119909⟩119860lowast (29)

Or 119879minus1minus2 le (119879lowast119879)minus1minus1 this implies

(10038171003817100381710038171003817119879minus110038171003817100381710038171003817minus1radic119898119860) ⟨119909 119909⟩ (10038171003817100381710038171003817119879minus1

10038171003817100381710038171003817minus1radic119898119860)lowast

le 119860⟨119870lowast119879119909119870lowast119879119909⟩119860lowast forall119909 isin 119880(30)

And we know that ⟨119879119909 119879119909⟩ le 1198792⟨119909 119909⟩ forall119909 isin 119880 Thisimplies that

119861 ⟨119879119909 119879119909⟩ 119861lowast le (119879 119861) ⟨119909 119909⟩ (119879 119861)lowast forall119909 isin 119880 (31)

Using (26) (30) (31) we have



le intΩ⟨Λ119908119879119909 Λ119908119879119909⟩ 119889120583 (119908)

le (119879 119861) ⟨119909 119909⟩ (119879 119861)lowast(32)

So Λ119908119879 119908 isin Ω is a continuous lowast-K-g-frame for 119880Moreover for every 119909 isin 119880 we have

119879lowast119878119879119909 = 119879lowast intΩΛlowast119908Λ119908119879119909119889120583 (119908)

= intΩ119879lowastΛlowast119908Λ119908119879119909119889120583 (119908)

= intΩ(Λ119908119879)lowast (Λ119908119879)119909119889120583 (119908)

(33)

This completes the proof

Corollary 10 Let Λ119908 isin 119864119899119889lowastA(119880119881119908) 119908 isin Ω bea continuous lowast-K-g-frame for 119880 and let 119870 isin 119864119899119889lowastA(119880)be surjective with continuous lowast-K-g-frame operator 119878 enΛ119908119878minus1 119908 isin Ω is a continuous lowast-K-g-frame for 119880Proof Result from the last theorem by taking 119879 = 119878minus1

The following theorem characterizes a continuous lowast-K-g-frame by its frame operator

Theorem 11 Let Λ 120596120596isinΩ be a continuous lowast-g-Bessel for 119867with respect to 119867120596120596isinΩ then Λ 120596120596isinΩ is a continuous lowast-K-g-frame for 119867 with respect to 119867120596120596isinΩ if and only if there existsa constant 119860 gt 0 such that 119878 ge 119860119870119870lowast where 119878 is the frameoperator for Λ 120596120596isinΩProof We know Λ 120596120596isinΩ is a continuous lowast-K-g-frame for119867with bounded 119860 and 119861 if and only if

119860⟨119870lowast119909 119870lowast119909⟩119860lowast le intΩ⟨Λ119908119909 Λ119908119909⟩ 119889120583 (119908)

le 119861 ⟨119909 119909⟩ 119861lowast(34)

If and only if

119860⟨119870119870lowast119909 119909⟩ 119860lowast le intΩ⟨Λlowast119908Λ119908119909 119909⟩ 119889120583 (119908)

le 119861 ⟨119909 119909⟩ 119861lowast(35)

If and only if

119860⟨119870119870lowast119909 119909⟩119860lowast le ⟨119878119909 119909⟩ le 119861 ⟨119909 119909⟩ 119861lowast (36)

where 119878 is the continuous lowast-K-g frame operator for Λ 120596120596isinΩTherefore the conclusion holds

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest


References

[1] R J Duffin andA C Schaeffer ldquoA class of nonharmonic fourierseriesrdquo Transactions of the American Mathematical Society vol72 no 2 pp 341ndash366 1952

[2] I Daubechies A Grossmann and Y Meyer ldquoPainless nonor-thogonal expansionsrdquo Journal of Mathematical Physics vol 27no 5 pp 1271ndash1283 1986

[3] D Gabor ldquoTheory of communicationsrdquo Journal of ElectricalEngineering vol 93 pp 429ndash457 1946

[4] G Kaiser A Friendly Guide to Wavelets Birkhauser BostonMass USA 1994

[5] S Ali J Antoine and J Gazeau ldquoContinuous frames in hilbertspacerdquo Annals of Physics vol 222 no 1 pp 1ndash37 1993

[6] J Gabardo and D Han ldquoFrames associated with measurablespacerdquo Advances in Computational Mathematics vol 18 no 3pp 127ndash147 2003

[7] A Askari-Hemmat M A Dehghan and M RadjabalipourldquoGeneralized frames and their redundancyrdquo Proceedings of theAmerican Mathematical Society vol 129 no 04 pp 1143ndash11482001

[8] M Rossafi and S Kabbaj ldquo-K-g-frames in Hilbert C-modulesrdquo Journal of Linear and Topological Algebra vol 7 no 1pp 63ndash71 2018

[9] K R Davidson C-Algebra by Example vol 6 of Fields InstituteMonographs American Mathematical Society Providence RI1996

[10] J B Conway A Course in Operator eory vol 21 of GraduateStudies in Mathematics American Mathematical Society 2000

[11] W L Paschke ldquoInner product modules over B-algebrasrdquoTransactions of the AmericanMathematical Society vol 182 pp443ndash468 1973

[12] L Arambasic ldquoOn frames for countably generated Hilbert Cmodulesrdquo Proceedings of the American Mathematical Societyvol 135 no 02 pp 469ndash479 2007

[13] A Alijani and M A Dehghan ldquo-frames in Hilbert C mod-ulesrdquo UPB Scientific Bulletin Series A vol 73 no 4 pp 89ndash1062011

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We reserve the notation 119864119899119889lowastA(HK) for the set of alladjointable operators from H to K and 119864119899119889lowastA(HH) isabbreviated to 119864119899119889lowastA(H)

The following lemmas will be used to prove our mainsresults

Lemma 2 (see [11]) Let H be Hilbert A-module If 119879 isin119864119899119889lowastA(H) then

⟨119879119909 119879119909⟩ le 1198792 ⟨119909 119909⟩ forall119909 isinH (1)

Lemma 3 (see [12]) Let H and K two Hilbert A-Modulesand 119879 isin 119864119899119889lowast(HK) en the following statements areequivalent

(i) 119879 is surjective(ii) 119879lowast is bounded below with respect to norm ie there is

119898 gt 0 such that 119879lowast119909 ge 119898119909 for all 119909 isinK(iii) 119879lowast is bounded below with respect to the inner product

ie there is 1198981015840 gt 0 such that ⟨119879lowast119909 119879lowast119909⟩ ge 1198981015840⟨119909 119909⟩for all 119909 isinK

Lemma 4 (see [13]) LetH andK be two HilbertA-Modulesand 119879 isin 119864119899119889lowast(HK) en

(i) if 119879 is injective and 119879 has closed range then theadjointable map 119879lowast119879 is invertible and

10038171003817100381710038171003817(119879lowast119879)minus110038171003817100381710038171003817minus1 119868H le 119879lowast119879 le 1198792 119868H (2)

(ii) If 119879 is surjective then the adjointable map 119879119879lowast isinvertible and

10038171003817100381710038171003817(119879119879lowast)minus110038171003817100381710038171003817minus1 119868K le 119879119879lowast le 1198792 119868K (3)

2 Continuous lowast-K-g-Frame in Hilbert119862lowast-Modules

Let119883be a Banach space (Ω 120583) ameasure space and function119891 Ω 997888rarr 119883 a measurable function Integral of the Banach-valued function 119891 has defined Bochner and others Mostproperties of this integral are similar to those of the integral ofreal-valued functions Because every 119862lowast-algebra and Hilbert119862lowast-module is a Banach space thus we can use this integraland its properties

Let (Ω 120583) be a measure space let119880 and 119881 be two Hilbert119862lowast-modules 119881119908 119908 isin Ω is a sequence of subspaces of Vand 119864119899119889lowastA(119880119881119908) is the collection of all adjointable A-linearmaps from119880 into 119881119908 We define

⨁119908isinΩ

119881119908

= 119909 = 119909119908 119909119908 isin 1198811199081003817100381710038171003817100381710038171003817intΩ

100381610038161003816100381611990911990810038161003816100381610038162 119889120583 (119908)1003817100381710038171003817100381710038171003817 lt infin

(4)

For any 119909 = 119909119908 119908 isin Ω and 119910 = 119910119908 119908 isinΩ if the A-valued inner product is defined by ⟨119909 119910⟩ =intΩ⟨119909119908 119910119908⟩119889120583(119908) the norm is defined by 119909 = ⟨119909 119909⟩12

the⨁119908isinΩ119881119908 is a Hilbert 119862lowast-module

Definition 5 Let119870 isin 119864119899119889lowastA(119880) we call Λ119908 isin 119864119899119889lowastA(119880119881119908) 119908 isin Ω a Continuous lowast-K-g-frame for Hilbert 119862lowast-module119880with respect to 119881119908 119908 isin Ω if

(a) for any 119909 isin 119880 the function 119909 Ω 997888rarr 119881119908 defined by119909(119908) = Λ119908119909 is measurable

(b) there exist two strictly nonzero elements 119860 and 119861 inA such that



The elements 119860 and 119861 are called Continuous lowast-K-g-framebounds

If 119860 = 119861 we call this Continuous lowast-K-g-frame a con-tinuous tight lowast-K-g-frame and if 119860 = 119861 = 1A it is calleda continuous Parseval lowast-K-g-frame If only the right-handinequality of (5) is satisfied we call Λ119908 119908 isin Ω acontinuous lowast-K-g-Bessel for 119880 with respect to Λ119908 119908 isin Ωwith Bessel bound 119861Example 6 Let 119897infin be the set of all bounded complex-valuedsequences For any 119906 = 119906119895119895isinN V = V119895119895isinN isin 119897infin we define

119906V = 119906119895V119895119895isinN 119906lowast = 119906119895119895isinN 119906 = sup

119895isinN

1003816100381610038161003816100381611990611989510038161003816100381610038161003816

(6)

ThenA = 119897infin is a Clowast-algebraLetH = 1198620 be the set of all sequences converging to zero

For any 119906 V isinH we define

⟨119906 V⟩ = 119906Vlowast = 119906119895119906119895119895isinN (7)

ThenH is a HilbertA-moduleDefine 119891119895 = 119891119895119894 119894isinNlowast by 119891119895119894 = 12 + 1119894 if 119894 = 119895 and 119891119895119894 = 0

if 119894 = 119895 forall119895 isin NlowastNow define the adjointable operator Λ 119895 H 997888rarr

A Λ 119895119909 = ⟨119909 119891119895⟩Then for every 119909 isinH we have

sum119895isinN⟨Λ 119895119909 Λ 119895119909⟩ = 12 +

1119894 119894isinNlowast ⟨119909 119909⟩

12 +

1119894 119894isinNlowast (8)

So Λ 119895119895 is a 12 + 1119894119894isinNlowast-tight lowast-g-frameLet119870 H 997888rarrH defined by 119870119909 = 119909119894119894119894isinNlowast Then for every 119909 isinH we have

⟨119870lowast119909119870lowast119909⟩Ale sum119895isinN⟨Λ 119895119909 Λ 119895119909⟩

= 12 +1119894 119894isinNlowast ⟨119909 119909⟩

12 +


(9)

Now let (Ω 120583) be a 120590-finite measure space with infinitemeasure and 119867120596120596isinΩ be a family ofHilbertA-module (119867120596 =1198620 forall119908 isin Ω)




Λ119908 (119909) = 1120583 (Ω119896) ⟨119909 119891119896⟩ ℎ120596 forall119909 isin 119867 (10)



intΩ⟨Λ 120596119909 Λ 120596119909⟩ 119889120583 (120596) = sum

119895isinN

⟨119909 119891119895⟩ ⟨119891119895 119909⟩ (11)

So therefore


= sum119895isinN

⟨119909 119891119895⟩ ⟨119891119895 119909⟩

= 12 +1119894 119894isinNlowast ⟨119909 119909⟩

12 +


(12)


Remark 7



forall119909 isin 119880(13)

or


then



(15)




119898⟨119909 119909⟩ le ⟨119870lowast119909119870lowast119909⟩ (16)

then




le 119861 ⟨119909 119909⟩ 119861lowast(18)








frame operator



⟨119878119909 119910⟩ = ⟨intΩΛlowast119908Λ119908119909119889120583 (119908) 119910⟩

= intΩ⟨Λlowast119908Λ119908119909 119910⟩ 119889120583 (119908)

= intΩ⟨119909 Λlowast119908Λ119908119910⟩ 119889120583 (119908)

= ⟨119909 intΩΛlowast119908Λ119908119910119889120583 (119908)⟩ = ⟨119909 119878119910⟩

(21)



le 119861 ⟨119909 119909⟩ 119861lowast(22)


So


This gives10038171003817100381710038171003817119860minus1



10038171003817100381710038171003817119860minus110038171003817100381710038171003817minus2 1198702 le 119878 le 1198612 (25)







119898⟨119879119909 119879119909⟩ le ⟨119870lowast119879119909119870lowast119879119909⟩ (27)

and then


so








Using (26) (30) (31) we have



le intΩ⟨Λ119908119879119909 Λ119908119879119909⟩ 119889120583 (119908)

le (119879 119861) ⟨119909 119909⟩ (119879 119861)lowast(32)




= intΩ(Λ119908119879)lowast (Λ119908119879)119909119889120583 (119908)

(33)






le 119861 ⟨119909 119909⟩ 119861lowast(34)

If and only if


le 119861 ⟨119909 119909⟩ 119861lowast(35)

If and only if



Data Availability





References





















Journal of












Journal of







Volume 2018






Volume 2018











Λ119908 (119909) = 1120583 (Ω119896) ⟨119909 119891119896⟩ ℎ120596 forall119909 isin 119867 (10)



intΩ⟨Λ 120596119909 Λ 120596119909⟩ 119889120583 (120596) = sum

119895isinN

⟨119909 119891119895⟩ ⟨119891119895 119909⟩ (11)

So therefore


= sum119895isinN

⟨119909 119891119895⟩ ⟨119891119895 119909⟩

= 12 +1119894 119894isinNlowast ⟨119909 119909⟩

12 +


(12)


Remark 7



forall119909 isin 119880(13)

or


then



(15)




119898⟨119909 119909⟩ le ⟨119870lowast119909119870lowast119909⟩ (16)

then




le 119861 ⟨119909 119909⟩ 119861lowast(18)








frame operator



⟨119878119909 119910⟩ = ⟨intΩΛlowast119908Λ119908119909119889120583 (119908) 119910⟩

= intΩ⟨Λlowast119908Λ119908119909 119910⟩ 119889120583 (119908)

= intΩ⟨119909 Λlowast119908Λ119908119910⟩ 119889120583 (119908)

= ⟨119909 intΩΛlowast119908Λ119908119910119889120583 (119908)⟩ = ⟨119909 119878119910⟩

(21)



le 119861 ⟨119909 119909⟩ 119861lowast(22)


So


This gives10038171003817100381710038171003817119860minus1



10038171003817100381710038171003817119860minus110038171003817100381710038171003817minus2 1198702 le 119878 le 1198612 (25)







119898⟨119879119909 119879119909⟩ le ⟨119870lowast119879119909119870lowast119879119909⟩ (27)

and then


so








Using (26) (30) (31) we have



le intΩ⟨Λ119908119879119909 Λ119908119879119909⟩ 119889120583 (119908)

le (119879 119861) ⟨119909 119909⟩ (119879 119861)lowast(32)




= intΩ(Λ119908119879)lowast (Λ119908119879)119909119889120583 (119908)

(33)






le 119861 ⟨119909 119909⟩ 119861lowast(34)

If and only if


le 119861 ⟨119909 119909⟩ 119861lowast(35)

If and only if



Data Availability





References





















Journal of












Journal of







Volume 2018






Volume 2018









So


This gives10038171003817100381710038171003817119860minus1



10038171003817100381710038171003817119860minus110038171003817100381710038171003817minus2 1198702 le 119878 le 1198612 (25)







119898⟨119879119909 119879119909⟩ le ⟨119870lowast119879119909119870lowast119879119909⟩ (27)

and then


so








Using (26) (30) (31) we have



le intΩ⟨Λ119908119879119909 Λ119908119879119909⟩ 119889120583 (119908)

le (119879 119861) ⟨119909 119909⟩ (119879 119861)lowast(32)




= intΩ(Λ119908119879)lowast (Λ119908119879)119909119889120583 (119908)

(33)






le 119861 ⟨119909 119909⟩ 119861lowast(34)

If and only if


le 119861 ⟨119909 119909⟩ 119861lowast(35)

If and only if



Data Availability





References





















Journal of












Journal of







Volume 2018






Volume 2018









References





















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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