Foguel-type operators similar to contractions

TL;DR

This paper characterizes seven types of Foguel-type operators, constructed from classical operators like shifts, Hankel, and Toeplitz, determining when they are similar to contractions.

Contribution

It provides a complete characterization of all seven Foguel-type operators that are similar to contractions, extending classical operator theory results.

Findings

Seven Foguel-type operators are fully characterized regarding their similarity to contractions.

The characterization involves classical operators such as shifts, Hankel, and Toeplitz operators.

The results generalize and unify previous understanding of operator similarity in this class.

Abstract

Pisier's celebrated counterexample to Halmos's similarity-to-contractions problem was based on $2 \times 2$ upper triangular block operator matrices involving three classical operators: forward and backward shifts on the diagonal and Hankel operators in the off-diagonal entry. Together with another classical object, namely Toeplitz operators, one can formulate another $2^3 -1 = 7$ types of $2 \times 2$ upper triangular block operator matrices, which we refer to as Foguel-type operators. In this paper, we give a complete characterization of all the seven Foguel-type operators being similar to contractions.