IUMJ

Title: Upper triangular Toeplitz matrices and real parts of quasinilpotent operators

Authors: Ken Dykema, Junsheng Fang and Anna Skripka

Issue: Volume 63 (2014), Issue 1, 53-75

Abstract:

We show that every self-adjoint matrix $B$ of trace $0$ can be realized as $B=T+T^{*}$ for a nilpotent matrix $T$ with $\|T\|\le K\|B\|$, for a constant $K$ that is independent of matrix size. More particularly, if $D$ is a diagonal, self-adjoint $n\times n$ matrix of trace $0$, then there is a unitary matrix $V=XU_n$, where $X$ is an $n\times n$ permutation matrix and $U_n$ is the $n\times n$ Fourier matrix, such that the upper triangular part, $T$, of the conjugate $V^{*}DV$ of $D$ satisfies $\|T\|\le K\|D\|$. This matrix $T$ is a strictly upper triangular Toeplitz matrix such that $T+T^{*}=V^{*}DV$. We apply this and related results to give partial answers to questions about real parts of quasinilpotent elements in finite von Neumann algebras.