On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum

TL;DR

This paper characterizes when bounded operators on a Hilbert space can be represented by specific three-diagonal matrices that are either complex symmetric or skew-symmetric with simple spectra, revealing their subset relations.

Contribution

It provides necessary and sufficient conditions for such matrix representations, advancing understanding of operator structures with simple spectra.

Findings

Operators with such matrices form a proper subset of all complex symmetric operators.

Similar conditions are established for complex skew-symmetric matrices.

Results clarify the structure of operators with simple spectra in Hilbert spaces.

Abstract

In this paper we obtain necessary and sufficient conditions for a linear bounded operator in a Hilbert space $H$ to have a three-diagonal complex symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in $H$. It is shown that a set of all such operators is a proper subset of a set of all complex symmetric operators with a simple spectrum. Similar necessary and sufficient conditions are obtained for a linear bounded operator in $H$ to have a three-diagonal complex skew-symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in $H$.