Invertibility of generalized Bessel multipliers in Hilbert $C^{*}$-modules
Bull. Korean Math. Soc. 2021 Vol. 58, No. 2, 461-479
Published online March 4, 2021
Printed March 31, 2021
Gholamreza Abbaspour Tabadkan, Hessam Hosseinnezhad
Damghan University; Damghan University
Abstract : This paper includes a general version of Bessel multipliers in Hilbert $C^*$-modules. In fact, by combining analysis, an operator on the standard Hilbert $C^*$-module and synthesis, we reach so-called generalized Bessel multipliers. Because of their importance for applications, we are interested to determine cases when generalized multipliers are invertible. We investigate some necessary or sufficient conditions for the invertibility of such operators and also we look at which perturbation of parameters preserve the invertibility of them. Subsequently, our attention is on how to express the inverse of an invertible generalized frame multiplier as a multiplier. In fact, we show that for all frames, the inverse of any invertible frame multiplier with an invertible symbol can always be represented as a multiplier with an invertible symbol and appropriate dual frames of the given ones.
Keywords : Bessel multiplier, modular Riesz basis, standard frame
MSC numbers : Primary 42C15, 46L08
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