Reducing subspaces of $C_{00}$ contractions

TL;DR

This paper investigates the reducibility of certain nonunitary contractions using Sz.-Nagy--Foias theory, applies findings to truncated Toeplitz operators, and refutes a previous conjecture in the field.

Contribution

It provides new general results on reducibility for a class of contractions and addresses a conjecture related to truncated Toeplitz operators.

Findings

Established conditions for reducibility of specific contractions

Applied results to truncated Toeplitz operators

Provided a negative answer to a prior conjecture

Abstract

Using the Sz.-Nagy--Foias theory of contractions, we obtain general results about reducibility for a class of completely nonunitary contractions. These are applied to certain truncated Toeplitz operators, previously considered by Li--Yang--Lu and Gu. In particular, a negative answer is given to a conjecture stated by the latter.