Invertibility of g-frame multipliers and Bessel multipliers for unitary systems in Hilbert C*-modules

ABSTRACT

In this paper, firstly we explore the invertibility of g-frame multipliers in Hilbert $C^{\ast }$-modules. We show that for ${\ell }^{\mathrm{\infty }}\left(\mathcal{A}\right)$-invertible symbols, a modular g-Riesz multiplier is automatically invertible, and that the inverse of any invertible g-frame multiplier can be represented as a g-frame multiplier and particularly, we determine a new case of invertible g-frame multipliers whose inverses are exact the g-frame multipliers with the inverse of the symbol and the canonical dual g-frames. A necessary and sufficient condition for the invertibility of g-frame multipliers is obtained from the operator-theoretic point of view, and we also show that a small perturbation is applied to the g-frame involved in an invertible g-frame multiplier can lead to a new invertible g-frame multiplier. We end the paper by introducing what we call Bessel multipliers for unitary systems in Hilbert $C^{\ast }$-modules and study their basic properties.

Keywords:

• Hilbert $C^{\ast }$-module
• g-frame multiplier
• modular g-Riesz multiplier
• unitary system

AMS SUBJECT CLASSIFICATIONS:

• 46L08
• 42B15
• 42C15
• 46H25

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The research is supported by the National Natural Science Foundation of China (Nos. 11761057 and 11561057).