Upper triangular Toeplitz matrices and real parts of quasinilpotent operators

TL;DR

This paper demonstrates how any trace-zero self-adjoint matrix can be expressed as the sum of a nilpotent matrix and its adjoint with controlled norm, and explores implications for quasinilpotent operators in von Neumann algebras.

Contribution

It provides a new construction linking self-adjoint matrices to nilpotent matrices via unitary conjugation, with norm bounds independent of matrix size.

Findings

Any trace-zero self-adjoint matrix can be decomposed into a nilpotent plus its adjoint with bounded norm.

Constructs explicit unitary matrices to realize the decomposition.

Provides partial answers regarding the structure of real parts of quasinilpotent elements.

Abstract

We show that every self--adjoint matrix B of trace 0 can be realized as B=T+T^* for a nilpotent matrix T of norm no greater than K times the norm of B, for a constant K that is independent of matrix size. More particularly, if D is a diagonal, self--adjoint n-by-n matrix of trace 0, then there is a unitary matrix V=XU_n, where X is an n-by-n permutation matrix and U_n is the n-by-n Fourier matrix, such that the upper triangular part, T, of the conjugate V^*DV of D has norm no greater than K times the norm of 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.