On the Schur–Horn problem

Abstract

Let $\mathcal{\mathscr{H}}$ be a separable Hilbert space. Recently, the concept of $K$-$g$-frame was introduced as a special generalization of $g$-Bessel sequences. In this paper, we point out some gaps in the proof of some existent results concerning $K$-$g$-frame. We present examples to indicate that these results are not necessarily valid. Then we remove the gaps and provide some desired conclusions. In this respect, we deal with Schur–Horn problem, which characterizes sequences ${\left\{\parallel f_n{\parallel }^2\right\}}_{n=1}^{\infty }$, for all frames ${\left\{f_n\right\}}_{n=1}^{\infty }$ with the same frame operator. We introduce the concept of synthesis related frames. Finally, as the main result, we investigate around Schur–Horn problem, for the case where $\mathcal{\mathscr{H}}$ is finite dimensional. In fact, we prove that two frames have the same frame operator if and only if they are synthesis related.