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 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.