Pairs of projections and commuting isometries

Abstract

It is known that the non-zero part of compact defect operators of Berger-Coburn-Lebow pairs (BCL pairs in short) of isometries are diagonal operators of the form \[ \begin{bmatrix} I_1 & & & \\ & D & & \\ & & - I_2 & \\ & & & - D \\ \end{bmatrix}, \] where $I_1$ and $I_2$ are the identity operators and $D$ is a positive contractive diagonal operator. We discuss the question of constructing an irreducible BCL pair from a diagonal operator of the above type. The answer to this question is sometimes in the affirmative and sometimes in the negative. This also answers a part of the question raised by He, Qin, and Yang. Our explicit constructions of BCL pairs yield concrete examples of pairs of commuting isometries.

28 pages, thoroughly revised. To appear in Journal of Operator Theory

Cited by in corpus (1)

• Berger-Coburn-Lebow representation for pure isometric representations of product system over ${\mathbb{N}}_0^2$