Frame theory is exciting and dynamic with applications to a wide variety of areas in mathematics and engineering. In this paper, we introduce the concept of Continuous ⁎-K-g-frame in Hilbert C^{⁎}-Modules and we give some properties.

1. Introduction and Preliminaries

The concept of frames in Hilbert spaces has been introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series, after the fundamental paper [2] by Daubechies, Grossman and Meyer, frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames [3].

Traditionally, frames have been used in signal processing, image processing, data compression, and sampling theory. A discreet frame is a countable family of elements in a separable Hilbert space which allows for a stable, not necessarily unique, decomposition of an arbitrary element into an expansion of the frame elements. The concept of a generalization of frames to a family indexed by some locally compact space endowed with a Radon measure was proposed by G. Kaiser [4] and independently by Ali, Antoine, and Gazeau [5]. These frames are known as continuous frames. Gabardo and Han in [6] called these frames associated with measurable spaces, Askari-Hemmat, Dehghan, and Radjabalipour in [7] called them generalized frames and in mathematical physics they are referred to as coherent states [5].

In this paper, we introduce the notion of Continuous ∗-K-g-Frame which are generalization of ∗-K-g-Frame in Hilbert C∗-Modules introduced by M. Rossafi and S. Kabbaj [8] and we establish some new results.

The paper is organized as follows: we continue this introductory section we briefly recall the definitions and basic properties of C∗-algebra, Hilbert C∗-modules. In Section 2, we introduce the Continuous ∗-K-g-Frame, the Continuous pre-∗-K-g-frame operator, and the Continuous ∗-K-g-frame operator; also we establish here properties.

In the following we briefly recall the definitions and basic properties of C∗-algebra, Hilbert A-modules. Our reference for C∗-algebras is [9, 10]. For a C∗-algebra A if a∈A is positive we write a≥0 and A+ denotes the set of positive elements of A.

Definition 1 (see [<xref ref-type="bibr" rid="B13">11</xref>]).

Let A be a unital C∗-algebra and H a left A-module, such that the linear structures of A and H are compatible. H is a pre-Hilbert A-module if H is equipped with an A-valued inner product 〈.,.〉A:H×H→A, such that is sesquilinear, positive definite, and respects the module action. In the other words,

〈x,x〉A≥0 for all x∈H and 〈x,x〉A=0 if and only if x=0.

〈ax+y,z〉A=a〈x,y〉A+〈y,z〉A for all a∈A and x,y,z∈H.

〈x,y〉A=〈y,x〉A∗ for all x,y∈H.

For x∈H, we define x=〈x,x〉A1/2. If H is complete with ., it is called a Hilbert A-module or a Hilbert C∗-module over A. For every a in C∗-algebra A, we have |a|=a∗a1/2 and the A-valued norm on H is defined by x=〈x,x〉A1/2 for x∈H.

Let H and K be two Hilbert A-modules. A map T:H→K is said to be adjointable if there exists a map T∗:K→H such that 〈Tx,y〉A=〈x,T∗y〉A for all x∈H and y∈K.

We reserve the notation EndA∗(H,K) for the set of all adjointable operators from H to K and EndA∗(H,H) is abbreviated to EndA∗(H).

The following lemmas will be used to prove our mains results

Lemma 2 (see [<xref ref-type="bibr" rid="B13">11</xref>]).

Let H be Hilbert A-module. If T∈EndA∗(H), then (1)Tx,Tx≤T2x,x,∀x∈H.

Lemma 3 (see [<xref ref-type="bibr" rid="B3">12</xref>]).

Let H and K two Hilbert A-Modules and T∈End∗(H,K). Then the following statements are equivalent:

T is surjective.

T∗ is bounded below with respect to norm; i.e., there is m>0 such that T∗x≥mx for all x∈K.

T∗ is bounded below with respect to the inner product; i.e., there is m′>0 such that 〈T∗x,T∗x〉≥m′〈x,x〉 for all x∈K.

Lemma 4 (see [<xref ref-type="bibr" rid="B4">13</xref>]).

Let H and K be two Hilbert A-Modules and T∈End∗(H,K). Then,

if T is injective and T has closed range, then the adjointable map T∗T is invertible and (2)T∗T-1-1IH≤T∗T≤T2IH.

If T is surjective, then the adjointable map TT∗ is invertible and (3)TT∗-1-1IK≤TT∗≤T2IK.

2. Continuous <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M95"><mml:mrow><mml:mi>∗</mml:mi></mml:mrow></mml:math></inline-formula>-K-<sc>g</sc>-Frame in Hilbert <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M96"><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>∗</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>-Modules

Let X be a Banach space, (Ω,μ) a measure space, and function f:Ω→X a measurable function. Integral of the Banach-valued function f has defined Bochner and others. Most properties of this integral are similar to those of the integral of real-valued functions. Because every C∗-algebra and Hilbert C∗-module is a Banach space thus we can use this integral and its properties.

Let (Ω,μ) be a measure space, let U and V be two Hilbert C∗-modules, {Vw:w∈Ω} is a sequence of subspaces of V, and EndA∗(U,Vw) is the collection of all adjointable A-linear maps from U into Vw. We define (4)⨁w∈ΩVw=x=xw:xw∈Vw,∫Ωxw2dμw<∞.For any x={xw:w∈Ω} and y={yw:w∈Ω}, if the A-valued inner product is defined by 〈x,y〉=∫Ω〈xw,yw〉dμ(w), the norm is defined by x=x,x1/2, the ⨁w∈ΩVw is a Hilbert C∗-module.

Definition 5.

Let K∈EndA∗(U); we call {Λw∈EndA∗(U,Vw):w∈Ω} a Continuous ∗-K-g-frame for Hilbert C∗-module U with respect to {Vw:w∈Ω} if

for any x∈U, the function x~:Ω→Vw defined by x~(w)=Λwx is measurable;

there exist two strictly nonzero elements A and B in A such that(5)AK∗x,K∗xA∗≤∫ΩΛwx,Λwxdμw≤Bx,xB∗,∀x∈U.

The elements A and B are called Continuous ∗-K-g-frame bounds.

If A=B we call this Continuous ∗-K-g-frame a continuous tight ∗-K-g-frame, and if A=B=1A it is called a continuous Parseval ∗-K-g-frame. If only the right-hand inequality of (5) is satisfied, we call {Λw:w∈Ω} a continuous ∗-K-g-Bessel for U with respect to {Λw:w∈Ω} with Bessel bound B.

Example 6.

Let l∞ be the set of all bounded complex-valued sequences. For any u=ujj∈N,v=vjj∈N∈l∞, we define (6)uv=ujvjj∈N,u∗=uj¯j∈N,u=supj∈Nuj.Then A={l∞,.} is a C∗-algebra.

Let H=C0 be the set of all sequences converging to zero. For any u,v∈H we define (7)u,v=uv∗=ujuj¯j∈N.Then H is a Hilbert A-module.

Define fj=fiji∈N∗ by fij=1/2+1/i if i=j and fij=0 if i≠j∀j∈N∗.

Now define the adjointable operator Λj:H→A,Λjx=〈x,fj〉.

Then for every x∈H we have (8)∑j∈NΛjx,Λjx=12+1ii∈N∗x,x12+1ii∈N∗.So Λjj is a 1/2+1/ii∈N∗-tight ∗-g-frame.

Let K:H→H defined by Kx=xi/ii∈N∗.

Then for every x∈H we have (9)K∗x,K∗xA≤∑j∈NΛjx,Λjx=12+1ii∈N∗x,x12+1ii∈N∗.

Now, let (Ω,μ) be a σ-finite measure space with infinite measure and Hωω∈Ω be a family of Hilbert A-module (Hω=C0,∀w∈Ω).

Since Ω is a σ-finite, it can be written as a disjoint union Ω=⋃Ωω of countably many subsets Ωω⊆Ω, such that μ(Ωk)<∞,∀k∈N. Without less of generality, assume that μ(Ωk)>0∀k∈N.

For each ω∈Ω, define the operator: Λω:H→Hw by (10)Λwx=1μΩkx,fkhω,∀x∈Hwhere k is such that w∈Ωω and hω is an arbitrary element of Hω, such that hω=1.

For each x∈H, Λωxω∈Ω is strongly measurable (since hω are fixed) and (11)∫ΩΛωx,Λωxdμω=∑j∈Nx,fjfj,x So, therefore (12)K∗x,K∗x≤∫ΩΛωx,Λωxdμω=∑j∈Nx,fjfj,x=12+1ii∈N∗x,x12+1ii∈N∗ So Λωω∈Ω is a continuous ∗-K-g-frame.

Remark 7.

Every continuous ∗-g-frame is a continuous ∗-K-g-frame indeed:

Let {Λw∈EndA∗(U,Vw):w∈Ω} be a continuous ∗-g-frame for Hilbert C∗-module U with respect to {Vw:w∈Ω}, then (13)Ax,xA∗≤∫ΩΛwx,Λwxdμw≤Bx,xB∗,∀x∈U.

or(14)K∗x,K∗x≤K2x,x,∀x∈U.

then(15)K-1AK∗x,K∗xK-1A∗≤∫ΩΛwx,Λwxdμw≤Bx,xB∗

so let {Λw∈EndA∗(U,Vw):w∈Ω} be a continuous ∗-K-g-frame with lower and upper bounds K-1A and B, respectively.

If K∈EndA∗(H) is a surjective operator, then every continuous ∗-K-g-frame for H with respect to {Vw:w∈Ω} is a continuous ∗-g-frame.

Indeed,

if K is surjective there exists m>0 such that (16)mx,x≤K∗x,K∗x

then (17)Amx,xAm∗≤AK∗x,K∗xA∗

or if {Λw∈EndA∗(U,Vw):w∈Ω} is a continuous ∗-K-g-frame, we have (18)Amx,xAm∗≤∫ΩΛwx,Λwxdμw≤Bx,xB∗

hence {Λw∈EndA∗(U,Vw):w∈Ω} is a continuous ∗-g-frame for U with lower and upper bounds Am and B, respectively

Let K∈EndA∗(U), and {Λw∈EndA∗(U,Vw):w∈Ω} be a continuous ∗-K-g-frame for Hilbert C∗-module U with respect to {Vw:w∈Ω}.

We define an operator T:U→⨁w∈ΩVw by(19)Tx=Λwx:w∈Ω∀x∈U,then T is called the continuous ∗-K-g-frame transform.

So its adjoint operator is T∗:⨁w∈ΩVw→U given by(20)T∗xωω∈Ω=∫ΩΛω∗xωdμwBy composing T and T∗, the frame operator S=T∗T given by

Sx=∫ΩΛω∗Λωxdμ(w), S is called continuous ∗-K-g frame operator

Theorem 8.

The continuous ∗-K-g frame operator S is bounded, positive, self-adjoint, and A-1-2K2≤S≤B2

Proof.

First we show, S is a self-adjoint operator. By definition we have ∀x,y∈U(21)Sx,y=∫ΩΛw∗Λwxdμw,y=∫ΩΛw∗Λwx,ydμw=∫Ωx,Λw∗Λwydμw=x,∫ΩΛw∗Λwydμw=x,Sy.Then S is a self-adjoint.

Clearly S is positive.

By definition of a continuous ∗-K-g-frame we have (22)AK∗x,K∗xA∗≤∫ΩΛwx,Λwxdμw≤Bx,xB∗.So(23)AK∗x,K∗xA∗≤Sx,x≤Bx,xB∗.This gives (24)A-1-2KK∗x,x≤Sx,x≤B2x,x.

If we take supremum on all x∈U, where x≤1, we have (25)A-1-2K2≤S≤B2.

Theorem 9.

Let K∈EndA∗(H) be surjective and {Λw∈EndA∗(U,Vw):w∈Ω} a continuous ∗-K-g-frame for U, with lower and upper bounds A and B, respectively, and with the continuous ∗-K-g-frame operator S.

Let T∈EndA∗(U) be invertible; then {ΛwT:w∈Ω} is a continuous ∗-K-g-frame for U with continuous ∗-K-g-frame operator T∗ST.

Proof.

We have(26)AK∗Tx,K∗TxA∗≤∫ΩΛwTx,ΛwTxdμw≤BTx,TxB∗,∀x∈U.Using Lemma 3, we have T∗T-1-1〈x,x〉≤〈Tx,Tx〉, ∀x∈U.

K is surjective, then there exists m such that (27)mTx,Tx≤K∗Tx,K∗Tx and then (28)mT∗T-1-1x,x≤K∗Tx,K∗Tx

so (29)mT∗T-1-1Ax,xA∗≤AK∗Tx,K∗TxA∗

Or T-1-2≤T∗T-1-1, this implies(30)T-1-1mAx,xT-1-1mA∗≤AK∗Tx,K∗TxA∗,∀x∈U.And we know that 〈Tx,Tx〉≤T2〈x,x〉, ∀x∈U. This implies that(31)BTx,TxB∗≤TBx,xTB∗,∀x∈U.Using (26), (30), (31) we have (32)T-1-1mAx,xT-1-1mA∗≤∫ΩΛwTx,ΛwTxdμw≤TBx,xTB∗

So {ΛwT:w∈Ω} is a continuous ∗-K-g-frame for U.

Moreover for every x∈U, we have (33)T∗STx=T∗∫ΩΛw∗ΛwTxdμw=∫ΩT∗Λw∗ΛwTxdμw=∫ΩΛwT∗ΛwTxdμw.This completes the proof.

Corollary 10.

Let {Λw∈EndA∗(U,Vw):w∈Ω} be a continuous ∗-K-g-frame for U and let K∈EndA∗(U) be surjective, with continuous ∗-K-g-frame operator S. Then {ΛwS-1:w∈Ω} is a continuous ∗-K-g-frame for U.

Proof.

Result from the last theorem by taking T=S-1

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

Theorem 11.

Let Λωω∈Ω be a continuous ∗-g-Bessel for H with respect to Hωω∈Ω, then Λωω∈Ω is a continuous ∗-K-g-frame for H with respect to Hωω∈Ω if and only if there exists a constant A>0 such that S≥AKK∗ where S is the frame operator for Λωω∈Ω.

Proof.

We know Λωω∈Ω is a continuous ∗-K-g-frame for H with bounded A and B if and only if (34)AK∗x,K∗xA∗≤∫ΩΛwx,Λwxdμw≤Bx,xB∗ If and only if (35)AKK∗x,xA∗≤∫ΩΛw∗Λwx,xdμw≤Bx,xB∗ If and only if (36)AKK∗x,xA∗≤Sx,x≤Bx,xB∗ where S is the continuous ∗-K-g frame operator for Λωω∈Ω.

Therefore, 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.

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