On the Schur-Horn problem. (English) Zbl 1472.42041

Abtahi, Fatemeh; Kamali, Zeinab; Keyshams, Zahra

Rocky Mt. J. Math. 51, No. 4, 1157-1170 (2021).

Summary: Let \(\mathcal{H}\) 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 \(\{\parallel f_n \parallel^2\}_{n = 1}^\infty \), for all frames \(\{f_n\}_{n = 1}^\infty\) 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 \(\mathcal{H}\) is finite dimensional. In fact, we prove that two frames have the same frame operator if and only if they are synthesis related.

MSC:

42C15	General harmonic expansions, frames
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Keywords:

frame; \(K\)-frame; \(K\)-\(g\)-frame; Schur-Horn problem; synthesis operator; unitary operator

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