Unitary equivalence to a truncated Toeplitz operator: analytic symbols

TL;DR

This paper develops criteria to determine when a matrix with distinct eigenvalues is unitarily equivalent to a truncated Toeplitz operator with an analytic symbol, providing a constructive test and exploring related operator equivalences.

Contribution

It introduces a constructive criterion for unitary equivalence to truncated Toeplitz operators with analytic symbols and proves all complex symmetric operators on small Hilbert spaces are such sums.

Findings

Provided a constructive test for unitary equivalence to truncated Toeplitz operators.

Proved all complex symmetric operators on Hilbert spaces of dimension ≤ 3 are sums of such operators.

Illustrated the criteria with several explicit examples.

Abstract

Unlike Toeplitz operators on $H^2$, truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this note we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension $\leq 3$ is unitarily equivalent to a direct sum of truncated Toeplitz operators.