• Title/Summary/Keyword: Hilbert $C^*$-modules

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THE WOVEN FRAME OF MULTIPLIERS IN HILBERT C* -MODULES

  • Irani, Mona Naroei;Nazari, Akbar
    • Communications of the Korean Mathematical Society
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    • v.36 no.2
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    • pp.257-266
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    • 2021
  • In this paper, by using the sequence of adjointable operators from C*-algebra 𝓐 into Hilbert 𝓐-module E, the woven frames of multipliers in Hilbert C*-modules are introduced. Meanwhile, we study the effect of operators on these frames and, also we construct the new woven frame of multipliers in Hilbert 𝓐-module 𝓐. Finally, compositions of woven frames of multipliers in Hilbert C*-modules are studied.

WOVEN g-FRAMES IN HILBERT C-MODULES

  • Rajput, Ekta;Sahu, Nabin Kumar;Mishra, Vishnu Narayan
    • Korean Journal of Mathematics
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    • v.29 no.1
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    • pp.41-55
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    • 2021
  • Woven frames are motivated from distributed signal processing with potential applications in wireless sensor networks. g-frames provide more choices on analyzing functions from the frame expansion coefficients. The objective of this paper is to introduce woven g-frames in Hilbert C∗-modules, and to develop its fundamental properties. In this investigation, we establish sufficient conditions under which two g-frames possess the weaving properties. We also investigate the sufficient conditions under which a family of g-frames possess weaving properties.

LINEAR MAPPINGS IN BANACH MODULES OVER A UNITAL C*-ALGEBRA

  • Lee, Jung Rye;Mo, Kap-Jong;Park, Choonkil
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.2
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    • pp.221-238
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    • 2011
  • We prove the Hyers-Ulam stability of generalized Jensen's equations in Banach modules over a unital $C^{\ast}$-algebra. It is applied to show the stability of generalized Jensen's equations in a Hilbert module over a unital $C^{\ast}$-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital $C^{\ast}$-algebra.

ON FRAMES FOR COUNTABLY GENERATED HILBERT MODULES OVER LOCALLY C*-ALGEBRAS

  • Alizadeh, Leila;Hassani, Mahmoud
    • Communications of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.527-533
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    • 2018
  • Let $\mathcal{X}$ be a countably generated Hilbert module over a locally $C^*$-algebra $\mathcal{A}$ in multiplier module M($\mathcal{X}$) of $\mathcal{X}$. We propose the necessary and sufficient condition such that a sequence $\{h_n:n{{\in}}\mathbb{N}\}$ in M($\mathcal{X}$) is a standard frame of multipliers in $\mathcal{X}$. We also show that if T in $b(L_{\mathcal{A}}(\mathcal{X}))$, the space of bounded maps in set of all adjointable maps on $\mathcal{X}$, is surjective and $\{h_n:n{{\in}}\mathbb{N}\}$ is a standard frame of multipliers in $\mathcal{X}$, then $\{T{\circ}h_n:n{\in}\mathbb{N}}$ is a standard frame of multipliers in $\mathcal{X}$, too.

A GENERALIZATION OF STONE'S THEOREM IN HILBERT $C^*$-MODULES

  • Amyari, Maryam;Chakoshi, Mahnaz
    • The Pure and Applied Mathematics
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    • v.18 no.1
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    • pp.31-39
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    • 2011
  • Stone's theorem states that "A bounded linear operator A is infinitesimal generator of a $C_0$-group of unitary operators on a Hilbert space H if and only if iA is self adjoint". In this paper we establish a generalization of Stone's theorem in the framework of Hilbert $C^*$-modules.

CONTROLLED K-FRAMES IN HILBERT C*-MODULES

  • Rajput, Ekta;Sahu, Nabin Kumar;Mishra, Vishnu Narayan
    • Korean Journal of Mathematics
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    • v.30 no.1
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    • pp.91-107
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    • 2022
  • Controlled frames have been the subject of interest because of their ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the notion of controlled K-frame or controlled operator frame in Hilbert C*-modules. We establish the equivalent condition for controlled K-frame. We investigate some operator theoretic characterizations of controlled K-frames and controlled Bessel sequences. Moreover, we establish the relationship between the K-frames and controlled K-frames. We also investigate the invariance of a controlled K-frame under a suitable map T. At the end, we prove a perturbation result for controlled K-frame.

THE GENERALIZED INVERSES A(1,2)T,S OF THE ADJOINTABLE OPERATORS ON THE HILBERT C^*-MODULES

  • Xu, Qingxiang;Zhang, Xiaobo
    • Journal of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.363-372
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    • 2010
  • In this paper, we introduce and study the generalized inverse $A^{(1,2)}_{T,S}$ with the prescribed range T and null space S of an adjointable operator A from one Hilbert $C^*$-module to another, and get some analogous results known for finite matrices over the complex field or associated rings, and the Hilbert space operators.