On the similarity of complex symmetric operators to perturbations of restrictions of normal operators

TL;DR

This paper investigates when complex symmetric operators can be considered similar to perturbations of restrictions of normal operators, providing specific results for a subclass of cyclic operators in finite dimensions.

Contribution

It establishes the similarity of certain cyclic complex symmetric operators to rank-one perturbations of restrictions of normal operators, using moment problems and spectral theory tools.

Findings

Proves similarity for a subclass of cyclic operators in finite dimensions.

Utilizes a truncated moment problem in complex analysis.

Connects spectral problems for Jacobi matrices to operator similarity.

Abstract

In this paper we consider a problem of the similarity of complex symmetric operators to perturbations of restrictions of normal operators. For a subclass of cyclic complex symmetric operators in a finite-dimensional Hilbert space we prove the similarity to rank-one perturbations of restrictions of normal operators. The main tools are a truncated moment problem in $\mathbb{C}$, and some objects similar to objects from the theory of spectral problems for Jacobi matrices.