In this article, we study tensor product of Hilbert $C^*$-modules and Hilbert spaces. We show that if $E$ is a Hilbert $A$-module and $F$ is a Hilbert $B$-module, then tensor product of frames (orthonormal bases) for $E$ and $F$ produce frames (orthonormal bases) for Hilbert $A\otimes B$-module $E\otimes F$, and we get more results.
For Hilbert spaces $H$ and $K$, we study tensor product of frames of subspaces for $H$ and $K$, tensor product of resolutions of the identities of $H$ and $K$, and tensor product of frame representations for $H$ and $K$.
Controlled $\ast$-K-fusion frames are generalization of controlled fusion frames in frame theory. In this paper, we propose the notion of controlled $\ast$-k-fusions frames on Hilbert $C^{\ast}$-modules. We give some caraterizations and some of their properties are obtained. Then we study the erasures of submodules of a controlled $k$-fusion frame in Hilbert $C^{\ast}$-modules and we present some sufficient conditions under which a sequence remains a standart controlled k-fusion frame after deletion of some submodules. Finally, we introduce a perturbation for controlled $K$-fusion frames in Hi…
In this paper, firstly we investigate conditions under which the action of an operator on a $K$-frame, remain again a $K$-frame for Hilbert module E. We also give a generalization of Douglas Theorem and we shall use it to prove the sum of two $K$-frame under certain condition is again a $K$-frame. Finally, we characterize the $K$-frame generators for a unitary system in terms of operators.
K-frames were introduced by L. Gavruta to study atomic systems on Hilbert spaces. Recently some generalizations of this concept are introduced and some of its difference with ordinary frames are studied. In this paper *-K-frames are introduced and some properties of this generalization of K-frames are studied. After proving some characterizations of *-K-frames, direct sum and tensor product of *-K-frames are considered and finally some perturbation results are established.
We introduce the concept of frame of multipliers in Hilbert modules over pro-C*-algebras and show that many properties of frames in Hilbert C*-modules are valid for frames of multipliers in Hilbert modules over pro-C*-algebras.
In this paper, we introduce controlled frames in Hilbert $C^*$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert $C^*$-module. Also multiplier operators for controlled frames in Hilbert $C^*$-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert $C^*$-modules and verify their relations to controlled frames and multiplier operators.
Frame theory has a great revolution in recent years. This new Theory have been extended from Hilbert spaces to Hilbert C*-modules. In this paper, we introduce the notion of dual *-K-g-frames in Hilbert A-modules. Lastly we study *-K-g-frames in tensor product of Hilbert C*-Modules and we establish some new results.
Frame Theory has a great revolution in recent years, this Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. The purpose of this paper is the introduction and the study of the concept of Controlled Continuous g-Frames in Hilbert $C^{\ast}$-Modules. Also we give some properties.
In this paper, we introduce the concept of Continuous $\ast$-g-Frame in Hilbert $C^{\ast}$-Modules and we establish some results. We also discuss the stability problem for Continuous $\ast$-g-Frame.
We generalize some results on compact operators on Hilbert spaces to "compact" operators on some Hilbert right W*-modules. We present in this frame the Schatten decomposition of the compact operators, the trace, the Banach Lp-spaces and their duals, the Hilbert-Schmitt operators, and the integral operators as an example of Hilbert-Schmitt operators.
We show that every infinite-dimensional commutative unital C*-algebra has a Hilbert C*-module admitting no frames. In particular, this shows that Kasparov's stabilization theorem for countably generated Hilbert C*-modules can not be extended to arbitrary Hilbert C*-modules.
In this paper, we study $\ast$-g-frames in tensor products of Hilbert $C^{\ast}$-modules. We show that a tensor product of two $\ast$-g-frames is a $\ast$-g-frames, and we get some result.
We introduce a new concept of frame operators for Banach spaces we call a Hilbert-Schauder frame operator. This is a hybird between standard frame theory for Hilbert spaces and Schauder frame theory for Banach spaces. Most of our results involve basic structure properties of the Hilbert-Schauder frame operator. Examples of Hilbert-Schauder frames include standard Hilbert frames and classical bases of $\ell_p$ and $L^p$-spaces with $1< p \le 2$. Finally, we give a new isomorphic characterization of Hilbert spaces.
Frames play significant role in various areas of science and engineering. In this paper, we introduce the concepts of frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H, K})$ and their generalizations. Moreover, we obtain some new results for generalized frames in Hilbert modules.
In this papers we investigate the g-frame and Bessel g-sequence related to a linear bounded operator $K$ in Hilbert $C^{\ast}$-module and we establish some results.
The goal of the present paper is to extend the theory of frames for countably generated Hilbert $C^*$-modules over arbitrary $C^*$-algebras. In investigating the non-unital case we introduce the concept of outer frame as a sequence in the multiplier module $M(X)$ that has the standard frame property when applied to elements of the ambient module $X$. Given a Hilbert $\A$-module $X$, we prove that there is a bijective correspondence of the set of all adjointable surjections from the generalized Hilbert space $\ell^2(\A)$ to $X$ and the set consisting of all both frames and outer frames for $X$.…
The paper gives analogues of some starting results in the theory of Gaussian Hilbert Spaces for semicircular distributed random variables. The transition from the commutative to the free frame is done considering matrices of increasing dimension and utilizing the Amitzur-Levitzki Theorem.
$K$-fusion frames are a generalization of fusion frames in frame theory. In this paper, we extend the concept of controlled fusion frames to controlled $K$-fusion frames, and we develop some results on the controlled $K$-fusion frames for Hilbert spaces, which generalized some well known of controlled fusion frames case. also we discuss some characterizations of controlled Bessel $K$-fusion sequences and of controlled Bessel $K$-fusion. Further, we analyse stability conditions of controlled $K$-fusion frames under perturbation.
In the paper we describe the C*-algebras of noncommutative spherical tight frames over some C*-algebras and then apply to study the noncommutative version of the universal classifying space.
Controlled frames have been the subject of interest because of its ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the concepts of controlled g-fusion frame and controlled K-g-fusion frame in Hilbert C*-modules and we give some properties. Also, we study the pertubation problem of controlled K-g-fusion frame. Moreover, an illustrative examples is presented to support the obtained results.
In this paper, we will generelize $b$-frames; a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product we will use a new product called the $b$-dual product and it is constructed via a bilinear mapping. We will introduce new results about this product, about $b$-frames, and about $K$-$b$-frames, and we will also give some examples of both $b$-frames and $K$-$b$-frames that have never been given before. We will give the expression of the reconstruction formul…
We investigate the structured frames for Hilbert $C^{*}$-modules. In the case that the underlying $C^{*}$-algebra is a commutative $W^*$-algebra, we prove that the set of the Parseval frame generators for a unitary operator group can be parameterized by the set of all the unitary operators in the double commutant of the group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also obtain the existence and uniquenes…
Woronowicz introduced the functional calculus for normal operators in Hilbert C*-modules. The aim of this paper is to translate, if possible, some basic properties of the functional calculus in Hilbert spaces to the Hilbert C*-module framework.
In the first part of the paper we describe the dual \ell^2(A)^{\prime} of the standard Hilbert C*-module \ell^2(A) over an arbitrary (not necessarily unital) C*-algebra A. When A is a von Neumann algebra, this enables us to construct explicitly a self-dual Hilbert A-module \ell^2_{\text{strong}}(A) that is isometrically isomorphic to \ell^2(A)^{\prime}, which contains \ell^2(A), and whose A-valued inner product extends the original inner product on \ell^2(A). This serves as a concrete realization of a general construction for Hilbert C*-modules over von Neumann algebras introduced by W. Paschk…
Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $\mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $\mathcal{H}_1$ which contains $\mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.
We construct an example of a Hilbert C*-module which shows that Troitsky's theorem on the geometrical essence of A-compact operators between Hilbert C*-modules is not extendable to a not countably generated module case (even in the case of a stronger uniform structure, which is also introduced). In addition, the constructed module admits no frames.
Regarding the generalizations of the Bessel inequality in Hilbert spaces which are due to Bombiari and Boas--Bellman, we obtain a version of the Bessel inequality and some generalizations of this inequality in the framework of Hilbert $C^*$-modules.
Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the …
Motivated from two decades old famous Feichtinger conjectures for frames, $R_\varepsilon$-conjecture and Weaver's conjecture for Hilbert spaces (and their solution by Marcus, Spielman, and Srivastava), we formulate Feichtinger conjectures for p-approximate Schauder frames, $R_\varepsilon$-conjecture, Weaver's conjectures and Akemann-Weaver conjectures for Banach spaces. We also formulate conjectures on p-approximate Schauder frames based on the results of Casazza for frames for Hilbert spaces. We state conjectures and problems for p-approximate Schauder frames based on fundamental inequality f…
