Journal of Function Spaces
Volume 2019 (2019), Article ID 2426978, 5 pages
https://doi.org/10.1155/2019/2426978

Research Article

Continuous ⁎-K-G-Frame in Hilbert -Modules

Mohamed Rossafi,¹ Abdeslam Touri,¹ Hatim Labrigui,¹ and Abdellatif Akhlidj²

¹Department of Mathematics, University of Ibn Tofail, B. P. 133, Kenitra, Morocco
²Department of Mathematics, University of Hassan II, Casablanca, Morocco

Correspondence should be addressed to Mohamed Rossafi; rossafimohamed@gmail.com

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

Academic Editor: Gestur Ólafsson

Copyright © 2019 Mohamed Rossafi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 -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 -algebra, Hilbert -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 -algebra, Hilbert -modules. Our reference for -algebras is [9, 10]. For a -algebra if is positive we write and denotes the set of positive elements of .

Definition 1 (see [11]). Let be a unital -algebra and a left -module, such that the linear structures of and are compatible. is a pre-Hilbert -module if is equipped with an -valued inner product , such that is sesquilinear, positive definite, and respects the module action. In the other words,

(i) for all and if and only if .(ii) for all and .(iii) for all .

For , we define . If is complete with , it is called a Hilbert -module or a Hilbert -module over . For every in -algebra , we have and the -valued norm on is defined by for .
Let and be two Hilbert -modules. A map is said to be adjointable if there exists a map such that for all and .
We reserve the notation for the set of all adjointable operators from to and is abbreviated to .

The following lemmas will be used to prove our mains results

Lemma 2 (see [11]). Let be Hilbert -module. If , then

Lemma 3 (see [12]). Let and two Hilbert -Modules and . Then the following statements are equivalent:

(i) is surjective.(ii) is bounded below with respect to norm; i.e., there is such that for all .(iii) is bounded below with respect to the inner product; i.e., there is such that for all .

Lemma 4 (see [13]). Let and be two Hilbert -Modules and . Then,

(i)if is injective and has closed range, then the adjointable map is invertible and

(ii)If is surjective, then the adjointable map is invertible and

2. Continuous -K-G-Frame in Hilbert -Modules

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

Let be a measure space, let and be two Hilbert -modules, is a sequence of subspaces of V, and is the collection of all adjointable -linear maps from into . We define

For any and , if the -valued inner product is defined by , the norm is defined by , the is a Hilbert -module.

Definition 5. Let ; we call a Continuous -K-g-frame for Hilbert -module with respect to if

(a)for any , the function defined by is measurable;(b)there exist two strictly nonzero elements and in such that

The elements and are called Continuous -K-g-frame bounds.
If we call this Continuous -K-g-frame a continuous tight -K-g-frame, and if it is called a continuous Parseval -K-g-frame. If only the right-hand inequality of (5) is satisfied, we call a continuous -K-g-Bessel for with respect to with Bessel bound .

Example 6. Let be the set of all bounded complex-valued sequences. For any , we define

Then is a -algebra.
Let be the set of all sequences converging to zero. For any we define

Then is a Hilbert -module.
Define by if and if .
Now define the adjointable operator .
Then for every we have

So is a -tight -g-frame.
Let defined by .
Then for every we have

Now, let be a -finite measure space with infinite measure and be a family of Hilbert -module .
Since is a -finite, it can be written as a disjoint union of countably many subsets , such that . Without less of generality, assume that .
For each , define the operator: by

where is such that and is an arbitrary element of , such that .
For each , is strongly measurable (since are fixed) and

So, therefore

So is a continuous -K-g-frame.

Remark 7.

(i)Every continuous -g-frame is a continuous -K-g-frame indeed: Let be a continuous -g-frame for Hilbert -module with respect to , then

 or

 then

 so let be a continuous -K-g-frame with lower and upper bounds and , respectively.(ii)If is a surjective operator, then every continuous -K-g-frame for with respect to is a continuous -g-frame. Indeed, if is surjective there exists such that

 then

 or if is a continuous -K-g-frame, we have

 hence is a continuous -g-frame for with lower and upper bounds and , respectively

Let , and be a continuous -K-g-frame for Hilbert -module with respect to .

We define an operator by

then is called the continuous -K-g-frame transform.

So its adjoint operator is given by

By composing and , the frame operator given by

, S is called continuous -K-g frame operator

Theorem 8. The continuous -K-g frame operator is bounded, positive, self-adjoint, and

Proof. First we show, is a self-adjoint operator. By definition we have

Then is a self-adjoint.
Clearly is positive.
By definition of a continuous -K-g-frame we have

So

This gives

If we take supremum on all , where , we have

Theorem 9. Let be surjective and a continuous -K-g-frame for , with lower and upper bounds and , respectively, and with the continuous -K-g-frame operator .
Let be invertible; then is a continuous -K-g-frame for with continuous -K-g-frame operator .

Proof. We have

Using Lemma 3, we have , .
is surjective, then there exists such that

and then

so

Or , this implies

And we know that , . This implies that

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

So is a continuous -K-g-frame for .
Moreover for every , we have

This completes the proof.

Corollary 10. Let be a continuous -K-g-frame for and let be surjective, with continuous -K-g-frame operator . Then is a continuous -K-g-frame for .

Proof. Result from the last theorem by taking

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

Theorem 11. Let be a continuous -g-Bessel for with respect to , then is a continuous -K-g-frame for with respect to if and only if there exists a constant such that where is the frame operator for .

Proof. We know is a continuous -K-g-frame for with bounded and if and only if

If and only if

If and only if

where is the continuous -K-g frame operator for .
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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