Diagonals of self-adjoint operators with finite spectrum (Q2787109)

The Schur-Horn theorem establishes that for finite sequences of nonincreasing real numbers \((\lambda_i)_{i=1}^N\) and \((d_i)_{i=1}^N\) the existence of a self-adjoint operator \(E:\mathbb K^N \to \mathbb K^N\) (for \(\mathbb K= \mathbb R\) or \(\mathbb C\)) with eigenvalues \((\lambda_i)\) and diagonal \((d_i)\) turns out to be equivalent to the majorization inequalities: \(\sum_{i=1}^N \lambda_i=\sum_{i=1}^N d_i\) and \(\sum_{i=1}^n \lambda_i\leq\sum_{i=1}^n d_i\) for \(1\leq n\leq N\).NEWLINENEWLINE Such a characterization in terms of majorization inequalities holds true also when dealing with positive compact operators acting on infinite dimensional Hilbert spaces. The main problem to face, even for diagonalizable operators, is the fact that the diagonal terms cannot be ordered when dealing with the situation of infinite sequences \((\lambda_i)\) and \((d_i)\). There are classes of operators where the problem can be overcome. For instance \textit{R. V. Kadison} gave a complete characterization of the diagonal sequences of orthogonal projections [Proc. Natl. Acad. Sci. USA 99, No. 7, 4178--4184 (2002; Zbl 1013.46049)] and [ibid. 99, No. 8, 5217--5222 (2002; Zbl 1013.46050)]. His results allows to handle the case of diagonal self-adjoint operators with two points in the spectrum. The three-point spectrum was previously considered by the second author [J. Funct. Anal. 265, No. 8, 1494--1521 (2013; Zbl 1301.47004)]. Also some partial solution in the case of normal operators with finite spectrum was achieved by \textit{W. Arveson} [Proc. Natl. Acad. Sci. USA 104, No. 4, 1152--1158 (2007; Zbl 1191.47027)]. The results in this paper can be seen as the analogue to the work by Arveson. A complete characterization is presented in the case of self-adjoint operators. Although it might be theoretically deduced from previous results by the authors [Trans. Am. Math. Soc. 367, No. 7, 5099--5140 (2015; Zbl 1316.42036)], the arguments given in this article are simpler than the ones presented in the mentioned paper and rely on Kadison's result and certain equivalences of Riemann majorization.