Continuous ⁎-K-G-Frame in Hilbert -Modules
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  in 1952 to study some deep problems in nonharmonic Fourier series, after the fundamental paper  by Daubechies, Grossman and Meyer, frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames .
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  and independently by Ali, Antoine, and Gazeau . These frames are known as continuous frames. Gabardo and Han in  called these frames associated with measurable spaces, Askari-Hemmat, Dehghan, and Radjabalipour in  called them generalized frames and in mathematical physics they are referred to as coherent states .
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  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 ). 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 ). Let be Hilbert -module. If , then
Lemma 3 (see ). 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 ). 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 thatThe 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 bythen is called the continuous -K-g-frame transform.
So its adjoint operator is given byBy 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 SoThis 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 haveUsing Lemma 3, we have , .
is surjective, then there exists such that and then so Or , this impliesAnd we know that , . This implies thatUsing (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.
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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