Continuous Frame in Hilbert $C^{\ast}$-Modules

Douglas' factorization theorem and atomic system in Hilbert pro$-C^{\ast}-$module

In the present paper we introduce the generalized inverse operators which have an interesting role in operator theory. We establish Douglas' factorization theorem type for Hilbert pro-$C^{\ast}$-module. We introduce the notion of atomic system and of $K$-frame in Hilbert pro-$C^{\ast}$-module and we study the relationship between them. We also demonstrat some properties of $K$-frame by using Douglas' factorization theorem.Finally we demonstrate that the sum of two $K$-frames in a Hilbert pro-$C^{\ast}$-module with certain conditions is once again a $K$-frame.

Hilbert $C^{\ast}-$模中的双框架

Biframes in Hilbert $C^{\ast}-$modules

本文引入Hilbert $C^{\ast}-$模中由序列对生成的双框架概念，提供实例，并从算子理论角度建立其性质以刻画特征。

希尔伯特$C^{\ast}-$模中的连续双框架

Continuous biframes in Hilbert $C^{\ast}-$modules

该研究引入希尔伯特$C^{\ast}-$模的连续双框架概念，借助可逆可伴算子刻画其特征，研究相关Bessel乘子与对偶框架，并拓展至张量积场景给出相关性质。

Controlled $\ast$-K-operator frame for $End_\mathcal{A}^\ast (\mathcal{H})$

Frame Theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper, we introduce the concept of Controlled $\ast$-$K$-operator frame for the space $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ of all adjointable operators on a Hilbert $\mathcal{A}$-module $\mathcal{H}$ and we establish some results.

Continuous $\ast$-g-Frame in Hilbert $C^{\ast}$-Modules

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.

$\ast$-g-frames in tensor products of hilbert $C^{\ast}$-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.

Continuous $\ast$-K-g-Frame in Hilbert $C^{\ast}$-Modules

In this paper, we introduce the concept of Continuous $\ast$-K-g-Frame in Hilbert $C^{\ast}$-Modules and we give some properties.

Controlled K-operator frame for $End_\mathcal{A}^\ast (\mathcal{H})$

Frame Theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper, we introduce the concept of Controlled K-operator frame for the space $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ of all adjointable operators on a Hilbert $\mathcal{A}$-module $\mathcal{H}$ and we establish some results.

$\ast$-Operator frame for $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$

In this Work, We introduce the concept of $\ast$-operator frame, which is a generalization of $\ast$-frames in Hilbert pro-$C^{\ast}$-modules, and we establish some results, we also study the tensor product of $\ast$-operator frame for Hilbert pro-$C^{\ast}$-modules.

Controlled continuous g-Frames in Hilbert $C^{\ast}$-Modules

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.

Hilbert $C^{\ast}$-模中连续 K-g-框架的构造

Construction of continuous K-g-Frames in Hilbert $C^{\ast}$-Modules

本文在Hilbert $C^{\ast}$-模中研究连续K-g-框架的构造与和，给出可共轭算子的充要条件以生成新框架，并讨论其和的特征。

Some generalizations of K-g-frames in Hilbert $C^{\ast}$- module

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.

Finite generation in $C^\ast$-algebras and Hilbert $C^\ast$-modules

We characterize $C^*$-algebras and $C^*$-modules such that every maximal right ideal (resp. right submodule) is algebraically finitely generated. In particular, $C^*$-algebras satisfy the Dales--Żelazko conjecture.

Hilbert $C^{\ast}$-模中的Bessel序列

Bessel sequences in Hilbert $C^{\ast}$-modules

本文研究Hilbert C*-模中Bessel序列的算子形式，提出紧致紧框架概念，并给出其编织理论。

Hilbert $C^{\ast}$-模中 $g$-框架的和

Sum of $g-$frames in Hilbert $C^{\ast}-$modules

研究 Hilbert $C^*$-模中 $g$-框架之和保持框架性质的条件，以相关算子可逆性为核心，并分析扰动稳定性。

Multipliers of Hilbert pro-C*-bimodules and crossed products by Hilbert pro-C*-bimodules

In this paper we introduce the notion of multiplier of a Hilbert pro-$C^{\ast }$-bimodule and we investigate the structure of the multiplier bimodule of a Hilbert pro-$C^{\ast}$-bimodule. We also investigate the relationship between the crossed product $A\times _{X}\mathbb{Z}$ of a pro-$% C^{\ast }$-algebra $A$ by a Hilbert pro-$C^{\ast }$-bimodule $X$ over $A$, the crossed product $M(A)\times _{M(X)}\mathbb{Z}$ of the multiplier algebra $M(A)$ of $A$ by the multiplier bimodule $M(X)$ of $X$ and the multiplier algebra $M(A\times _{X}\mathbb{Z})$ of $A\times _{X}\mathbb{Z}$.

Operator frame for $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$

The concept of operator frame can be considered as a generalization of frame. Firstly, we introduce the notion of operator frame for the set of all adjointable operators $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$ on a Hilbert pro-$C^{\ast}$-module $\mathcal{X}.$ The analysis operator, the synthesis operator and the frame operator are presented. Secondly, we study the stability of operator frame under small perturbations. We also study the tensor product of operator frame for Hilbert pro-$C^{\ast}$-modules. Finally, we establish its dual and some properties.

Hilbert spaces and ${C}^\ast$-algebras are not finitely concrete

We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an analogous result for the category of commutative unital $C^\ast$-algebras with $\ast$-homomorphisms. This implies, in particular, that this category is not axiomatizable by a first-order theory, a strengthening of a conjecture of Bankston.

Some inequalities for $(α, β)$-normal operators in Hilbert spaces

An operator $T$ acting on a Hilbert space is called $(α,β)$-normal ($0\leq α\leq 1\leq β$) if \begin{equation*} α^{2}T^{\ast }T\leq TT^{\ast}\leq β^{2}T^{\ast}T. \end{equation*} In this paper we establish various inequalities between the operator norm and its numerical radius of $(α,β)$-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces.

Hilbert $C^{\ast}$-模空间中两个算子的欧几里得算子半径的若干不等式

Some inequalities for the Euclidean operator radius of two operators in Hilbert $C^{\ast}$-Modules space

本文研究Hilbert $C^*$-模中两个有界线性算子的欧几里得算子半径，给出若干精确界，并将其与已有数值半径上下界结果建立联系。

Hilbert C*-模上同态与*-同态的Hyers-Ulam稳定性

Hyers-Ulam stability of homomorphisms and *-homomorphisms on Hilbert C*-modules

该研究提出Hilbert C*-模上*-同态的概念，并采用不动点方法证明了此类模上同态与*-同态的Hyers-Ulam稳定性。

Hilbert C*-模中g-融合框架的表示

Representations of $g$-fusion frames in Hilbert $C^{\ast}$-Modules

本文在Hilbert C*-模中推广融合框架概念，研究其通过线性算子的可表示性，证明表示算子的有界性、紧g-融合框架不可逆表示性，并讨论扰动表示与稳定性。

On the ternary domain of a completely positive map on a Hilbert C*-module

We associate to an operator valued completely positive linear map $φ$ on a $C^{\ast }$-algebra $A$ and a Hilbert $C^{\ast }$-module $X$ over $A$ a subset $X_{φ}$ of $X,$ called '\textit{ternary domain}' of $φ$ on $X,$ which is a Hilbert $C^{\ast }$-module over the multiplicative domain of $φ$ and every $φ$-map (i.e., associated quaternary map with $φ$) acts on it as a ternary map. We also provide several characterizations for this set. The ternary domain \ of $φ$ on $A\ $ is a closed two-sided $\ast $-ideal $T_{φ}$ of the multiplicative domain of $φ$. We show that $XT_{φ}=X_{φ}$ and give sever…

Integral $K$-Operator Frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$

In this work, we introduce a new concept of integral $K$-operator frame for the set of all adjointable operators from Hilbert $C^{\ast}$-modules $\mathcal{H}$ to it self noted $End_{\mathcal{A}}^{\ast}(\mathcal{H}) $. We give some propertis relating some construction of integral $K$-operator frame and operators preserving integral $K$-operator frame and we establish some new results.

The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups on $ C^{\ast} $-Algebras of Compact Operators

In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every $ C^{\ast} $-dynamical system of the form $ (G,\mathbb{K}(\mathcal{H}),α) $, where $ G $ is a locally compact Hausdorff abelian group and $ \mathcal{H} $ is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert $ \mathbb{K}(\mathcal{H}) $-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hil…

Some generalizations of frames in Hilbert modules

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.

Frames in Quaternionic Hilbert Spaces

In this paper, we introduce and study the frames in separable quaternionic Hilbert spaces. Results on the existence of frames in quaternionic Hilbert spaces have been given. Also, a characterization of frame in quaternionic Hilbert spaces in terms of frame operator is given. Finally, a Paley-Wiener type perturbation result for frames in quaternionic Hilbert space has been obtained.

$\ast$-operator frame for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$

In this paper, a new notion of frames is introduced: $\ast$-operator frame as generalization of $\ast$-frames in Hilbert $C^{\ast}$-modules introduced by A. Alijani and M. A. Dehghan \cite{Ali} and we establish some results.

Frame of Operators in quaternionic Hilbert spaces

In this paper, we introduce and study frame of operators in quaternionic Hilbert spaces as a generalization of g frames which in turn generalized various notions like Pseduo frames, bounded quasi-projectors and frame of subspaces (fusion frames) in separable quaternionic Hilbert spaces.

The duals of $\ast$-operator frames for $End_{\mathcal{A}}^{\ast}(H)$

Frames play significant role in signal and image processing, which leads to many applications in differents fields. In this paper we define the dual of $\ast$-operator frames and we show their propreties obtained in Hilbert $\mathcal{A}$-modules and we establish some results.

Continuous relay fusion frame in Hilbert spaces

In this paper we introduced the concept of continuous relay fusion frames in Hilbert spaces. And we define the dual frames for continuous relay fusion frames. Finally we study the perturbation probleme of continuous relay fusion frames.

Construction of frame relative to n-Hilbert space

In this paper, our aim is to introduce the concept of a frame in n-Hilbert space and describe some of their properties. We further discuss tight frame relative to n-Hilbert space. At the end, we study the relationship between frame and bounded linear operator in n-Hilbert space.

Tight projections of frames on infinite dimensional Hilbert spaces

We characterize the frames on an infinite dimensional separable Hilbert space that can be projected to a tight frame for an infinite dimensional subspace. A result of Casazza and Leon states that an arbitrary frame for a 2N- or (2N-1)-dimensional Hilbert space can be projected to a tight frame for an N-dimensional subspace. Surprisingly, we demonstrate a large class of frames for infinite dimensional Hilbert spaces which cannot be projected to a tight frame for any infinite dimensional subspace.

More sums of Hilbert frames

In this paper we have some new results on sums of Hilbert space frames and Riesz bases. We also have a correction for some results in "S. Obeidat et al., Sums of Hilbert space frames, J. Math. Anal. Appl. 351 (2009) 579-585."

Duals of a frame in quaternionic Hilbert spaces

Frames in a separable quaternionic Hilbert space were introduced and studied in [17] to have more applications. In this paper, we extend the study of frames in quaternionic Hilbert spaces and introduce different types of duals of a frame in separable quaternionic Hilbert spaces. As an application, we give the orthogonal projection of $\ell^2(\HH)$ onto the range of analysis operator of the given frame, in terms of elements of canonical dual frame and elements of the frame in quaternionic Hilbert space. Finally, we give an expression for the orthogonal projection in terms of operators related t…

Generalized fusion frame in tensor product of Hilbert spaces

Generalized fusion frame and some of their properties in tensor product of Hilbert spaces are described. Also, the canonical dual g-fusion frame in tensor product of Hilbert spaces is considered. Finally, the frame operator for a pair of g-fusion Bessel sequences in tensor product of Hilbert spaces is presented.

Some properties of K-frame in n-Hilbert space

The notion of a K-frame in n-Hilbert space is presented and some of their characterizations are given. We verify that sum of two K-frames is also a K-frame in n-Hilbert space. Also, the concept of tight K-frame in n-Hilbert space is described and some properties of its are going to be established.

四元数 Hilbert 空间中的广义融合框架

Generalized fusion frame in Quaternionic Hilbert spaces

本文研究四元数 Hilbert 空间中的广义融合框架，给出其基于框架算子的刻画，并利用可逆有界右 Q-线性算子构造 g-fusion frame。

n-希尔伯特空间张量积中的框架引论

Introduction of frame in tensor product of n-Hilbert spaces

本研究针对n-希尔伯特空间张量积仍为n-希尔伯特空间的特性，引入框架概念，推广基的相关结果，探究框架与有界线性算子的关系，并讨论对偶框架，拓展了n-希尔伯特空间框架理论的研究范畴。

Continuous frames in Hilbert C*-modules

In the present paper the notion of continuous frames is introduced and some results of these frames are proved. Next, we give the concept of duals of continuous frames in Hilbert C*-modules and investigate some properties of them.

n-希尔伯特空间中的连续框架及其张量积

Continuous frames in n-Hilbert spaces and their tensor products

该研究提出n-希尔伯特空间中连续框架的概念，将其推广至n-希尔伯特空间的张量积场景，并进一步研究了对应空间及张量积下的对偶连续框架与连续Bessel乘子。

Kaczmarz algorithm in Hilbert space and tight frames

We prove that any tight frame in Hilbert space can be obtained by the Kaczmarz algorithm. An explicit way of constructing this correspondence is given. The uniqueness of the correspondence is determined.

Approximation of the Inverse Frame Operator and Stability of Hilbert$-$Schmidt Frames

In this paper, we study the Hilbert$-$Schmidt frame (HS-frame) theory for separable Hilbert spaces. We first present some characterizations of HS-frames and prove that HS-frames share many important properties with frames. Then, we show how the inverse of the HS-frame operator can be approximated using finite-dimensional methods. Finally, we present a classical perturbation result and prove that HS-frames are stable under small perturbations.

Dual pair and Approximate dual for continuous frames in Hilbert spaces

In this manuscript, the concept of dual and approximate dual for continuous frames in Hilbert spaces will be introduced. Some of its properties will be studied. Also, the relations between two continuous Riesz bases in Hilbert spaces will be clarified through examples.

希尔伯特空间中bi-g-fusion框架的构造

Constructions of bi-g-fusion frame in Hilbert space

该研究提出了希尔伯特空间中作为受控g-fusion框架推广的bi-g-fusion框架概念，给出示例并探讨了其在希尔伯特空间张量积中的情况。

Dual continuous $K$-Frames in Hilbert spaces

Frame theory is recently an active research area in mathematics, computer science and engineering with many exciting applications in a variety of different fields. This theory has been generalized rapidly and various generalizations of frames in Hilbert spaces. In this papers we study the notion of dual continuous $K$-frames in Hilbert spaces. Also we etablish some new properties.

Controlled frames in n-Hilbert spaces and their tensor products

The concepts of controlled frames and it's dual in n-Hilbert spaces and their tensor products have been introduced and then some of their characterizations are given. We further study the relationship between controlled frame and bounded linear operator in tensor product of n-Hilbert spaces. At the end, the direct sum of controlled frames in n-Hilbert space is being considered.

Local theory of frames and Schauder bases for Hilbert space

We develope a local theory for frames on finite dimensional Hilbert spaces. In particular, a bounded frame on a finite dimensional Hilbert space contains a subset which is a good Riesz basis for a percentage (arbitrarily close to one) of the space. We also construct a normalized frame for a Hilbert space which contains a subset which is a Schauder basis for H but does not contain any subset which is a Riesz basis for H.

Nagy和Pollard-Hilding型Hilbert-Schmidt框架的稳定性

Nagy and Pollard-Hilding-Type Stability of Hilbert-Schmidt Frames

本文为可分Hilbert空间中Hilbert-Schmidt框架的扰动稳定性建立了基于Nagy型和Pollard-Hilding型不等式的充分条件，并提供了框架界的显式估计及示例。

Continuous controlled generalized fusion frames in Hilbert spaces

We introduce the notion of continuous controlled g-fusion frame in Hilbert space which is the generalization of discrete controlled g-fusion frame and give an example. Some characterizations of continuous controlled g-fusion frame have been presented. We define the frame operator and multiplier of continuous controlled g-fusion Bessel families in Hilbert spaces. Continuous resolution of the identity operator on a Hilbert space using the theory of continuous controlled g-fusion frame is being considered. Finally, we discuss perturbation results of continuous controlled g-fusion frame.

Controlled weaving frames in Hilbert spaces

In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving K-g-frames in separable Hilbert spaces. Also, a characterization of controlled weaving K-g-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled K-g-woven frames gives ordinary weaving K-frames and vice-versa.

希尔伯特空间张量积中的融合框架及其替代对偶

Fusion frame and its alternative dual in tensor product of Hilbert spaces

该研究聚焦希尔伯特空间张量积中的融合框架，分析其相关性质与恒等算子的单位分解，并提出该场景下融合框架的替代对偶，丰富了泛函分析中融合框架的理论体系。

Generalized fusion frames in Hilbert spaces

After introducing g-frames and fusion frames by Sun and Casazza, combining these frames together is an interesting topic for research. In this paper, we introduce the generalized fusion frames or g-fusion frames for Hilbert spaces and give characterizations of these frames from the viewpoint of closed range and g-fusion frame sequences. Also, the canonical dual g-fusion frames are presented and we introduce Parseval g-fusion frames.

Controlled continuous $g$-frames and their duals in Hilbert spaces

In this paper, we characterize and study the concept of controlled continuous $g$-frame which is an extension of continuous $g$-frame in Hilbert spaces. We introduce the concept of controlled continuous dual $g$-frame and observe some interesting properties of it. Finally, we characterize all controlled continuous dual $g$-frames of a controlled continuous $g$-frame.