Invertibility of generalized g-frame multipliers in Hilbert spaces

ABSTRACT

In this paper, we investigate the invertibility of generalized g-Bessel multipliers. Sufficient and necessary conditions for invertibility are determined depending on the optimal g-frame bounds. Moreover, we show that, for semi-normalized symbols, the inverse of any invertible generalized g-frame multiplier can be represented as a generalized g-frame multiplier with the reciprocal symbol and dual g-frames of the given ones. Furthermore, we investigate some equivalent conditions for the special case, when both dual g-frames can be chosen to be the canonical duals. Finally, we give several approaches for constructing invertible generalized g-frame multipliers from the given ones. It is worth mentioning that some of our results are quite different from those studied in the previous literatures on this topic.

Keywords:

• G-Bessel sequences
• g-frames
• generalized multipliers
• perturbation

2010 Mathematics Subject Classifications:

• 42C15
• 47A05
• 41A58

Acknowledgments

The research was supported by Razi University of Kermanshah. The authors would like to thank this university for its kind supports. The authors are grateful to the referee for careful reading of the paper and for suggestions. In particular, Section 3 has been added to the article by the one suggested by him/her.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was supported by Razi University of Kermanshah.