hilbert $c^*$

The present paper considers duals‎, ‎approximate duals and pseudo-duals of generalized frames in Hilbert C*-modules‎. ‎In particular‎, ‎the ones constructed by bounded operators inserted between the synthesis and analysis operators of a Bessel sequence are focused and characterized‎. ‎Moreover‎, ‎the mentioned notions for modular g-Riesz bases are studied and some of their properties are obtained.

We shall provide a brief account on new achievements in the study of composition and composition-differentiation operators acting on classical spaces of analytic functions on the unit disk, as well as on the polydisk and the unit ball. The emphasis is given to the Hardy space, the Bergman space, and the Dirichlet space on the unit disk in complex plane. In the last section, we shall provide some new results regarding the Hilbert-Schmidt operators in the setting of several complex variables.

In 2016, Bemrose et al. introduced the weaving frames in a Hilbert space which is influenced by a problem in distributed signal processing. Ghobadzadeh et al. proposed the idea of woven frames in Hilbert $C^*$-modules in 2018. The authors studied and investigated numerous elementary properties of weaving frames in Hilbert $C^*$-modules. As K-frames and standard frames deviate in several perspectives, we acquaint the notion of weaving K-frames and an atomic system for weaving K-frames in Hilbert $C^*$-modules. Inside this script, we explore weaving K-frames from an operator theoretic point of view. We provide an identical interpretation for weaving K-frames and characterize weaving K-frames in terms of bounded linear operators. We also inspect the invariance of woven Bessel sequences under an adjointable operator.

In the present article, duals, approximate duals and pseudo-duals (generated by bounded and not necessarily adjointable operators) of a frame in a Hilbert $C^\ast$-module are characterized and some of their properties are obtained. Especially, the ones constructed by multiplication operators are discussed and their stability under the action of morphisms is focused and some equivalent conditions for the stability are derived. Finally, we get some results on pseudo-duals of modular Riesz bases, mainly their preservation under the action of morphisms and their behavior in the presence of semi-normalized symbols are studied.

In the present paper, we introduce  the generalized inverse operators, which have an exciting role in operator theory. We establish Douglas' factorization theorem type  for  the Hilbert pro-$C^{\ast}$-module.We introduce the notion of atomic system and $K$-frame in the Hilbert pro-$C^{\ast}$-module and study their relationship. We also demonstrate some properties of the $K$-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.

In the present paper, we study continuous frames in Hilbert $C^*$-modules and present some results of these frames. Next, we give the concept of dual continuous frames in Hilbert $C^*$-modules and investigate some properties of them. Also, by introducing the notion of the similarity of the continuous frames, characterizing it, and stating some of its properties, we refer to the investigation of the effect of similarity on the dual continuous frames in Hilbert $C^*$-modules.

The paper is devoted to continuous frames and continuous Riesz basis in Hilbert C*-modules. We define a continuous Riesz basis in Hilbert C*-modules and investigate the relationship between a continuous Riesz basis and an L^2-independent Bessel mapping. Also, we show that a continuous frame is a continuous Riesz basis if and only if it is a Riesz-type frame. Finally, we give the relation between two continuous Riesz bases in Hilbert C*-modules.

‎In this note‎ , ‎we present some operator ‎ine‎qualities via convexity property‎. ‎In the end‎, ‎refinements of mixed mean inequalities are given‎.

In this paper, we introduce the notion of continuous ∗-frames and ∗-C-controlled frames in Hilbert C∗-modules. We present some results of continuous frames in the view of ∗-C-controlled frames in Hilbert C∗-modules. Also we define ∗-(C, C')-frames and investigate multiplier operators for these frames.

Let $ \mathcal{A} $ be a standard $ C^{*} $-algebra. In this paper, we will study the continuity of $ \varepsilon $-orthogonality preserving mappings between Hilbert $ \mathcal{A} $-modules. Moreover, we will show that a local mapping between Hilbert $ \mathcal A $-modules is  $ \mathcal A $-linear. Furthermore, we will prove that for a pair of nonzero $ \mathcal A $-linear mappings $ T,S:E\longrightarrow F $, between Hilbert  $ \mathcal A $-modules, satisfying  $ \varepsilon $-orthogonality preserving property, there exists $ \gamma \in \mathbb{C} $,\begin{align*}    \Vert \langle T(x), S(y) \rangle-\gamma \langle x, y \rangle\Vert \leq \varepsilon \Vert T\Vert \Vert S\Vert \Vert x\Vert \Vert y \Vert,  \quad x, y \in E.\end{align*}Our results generalize the known ones in the context of Hilbert spaces.

Let Hμ=(μn+k)n,k≥0 with entries μn,k=μn+k induces the operator Hμ(f)(z)=∑∞n=0(∑∞k=0μn,kak)zn on the space of all analytic functions f(z)=∑∞n=0anzn in the unit disk D, where μ is a positive Borel measure on the interval [0,1). In this paper, we characterize the boundedness and compactness of the operator Hμ on Zygmund type spaces.

The aim of this paper is to investigate the closed-ness of the range of the modular operators in Hilbert C*-modules. We present the conditions that, the reverse order law for the closed range modular operators and modular projections be holds. Also, we prove that, for modular operators A and B with closed ranges, if BA=0 then A†B† =0. Moreover, we give a novel characterization of the normal modular operators in Hilbert C*-modules.

In this paper, we study the theory of dilation in operator algebras by introducing theorems that have the same theme. We propose some important theorems and point out the existence of the same proof idea for such theorems.