Pairs of projections and commuting isometries

Sandipan De, Shankar. P, Jaydeb Sarkar, Sankar T. R

Published 2020-08-27Version 1

Given a separable Hilbert space $\mathcal{E}$, let $H^2_{\mathcal{E}}(\mathbb{D})$ denote the $\mathcal{E}$-valued Hardy space on the open unit disc $\mathbb{D}$ and let $M_z$ denote the shift operator on $H^2_{\mathcal{E}}(\mathbb{D})$. It is known that a commuting pair of isometries $(V_1, V_2)$ on $H^2_{\mathcal{E}}(\mathbb{D})$ with $V_1 V_2 = M_z$ is associated to an orthogonal projection $P$ and a unitary $U$ on $\mathcal{E}$ (and vice versa). In this case, the ``defect operator'' of $(V_1, V_2)$, say $T$, is given by the difference of orthogonal projections on $\mathcal{E}$: \[ T = UPU^* - P. \] This paper is an attempt to determine whether irreducible commuting pairs of isometries $(V_1, V_2)$ can be built up from compact operators $T$ on $\mathcal{E}$ such that $T$ is a difference of two orthogonal projections. The answer to this question is sometimes in the affirmative and sometimes in the negative. \noindent The range of constructions of $(V_1, V_2)$ presented here also yields examples of a number of concrete pairs of commuting isometries.