# On some equalities and inequalities for $K$-frames

## TL;DR

This paper extends known equalities and inequalities from ordinary frames to K-frames in Hilbert spaces, providing new mathematical tools relevant for sampling theory and operator analysis.

## Contribution

It introduces several new equalities and inequalities for K-frames, expanding the theoretical framework and applying Jensen's operator inequality for further results.

## Key findings

- New equalities for K-frames established
- Novel inequalities derived using Jensen's operator inequality
- Enhanced understanding of K-frame properties in Hilbert spaces

## Abstract

K-frame theory was recently introduced to reconstruct elements from the range of a bounded linear operator K in a separable Hilbert space. This significant property is worthwhile especially in some problems arising in sampling theory. Some equalities and inequalities have been established for ordinary frames and their duals. In this paper, we continue and extend these results to obtain several important equalities and inequalities for K-frames. Moreover, by applying Jensen's operator inequality we obtain some new inequalities for $K$-frames.