Algebraic properties of Truncated Toeplitz $+$ Truncated Hankel operators

Abstract

In this paper, we study the algebraic properties of ${\mathcal{S}}_{\theta }$---the class of truncated Toeplitz plus truncated Hankel operators --- on the model space ${\mathcal{K}}_{\theta }$. We also give a complete characterization of the intersection class of truncated Toeplitz and truncated Hankel operators under certain conditions. As applications, we describe the non-uniqueness of a decomposition $T=A+B$, show that self-adjoint operators admit self-adjoint decompositions, give a necessary and sufficient inclusion criterion for ${\mathcal{S}}_{\theta }$ when $\theta$ is a finite Blaschke product, and, from a first-order commutator characterization of ${\mathcal{S}}_{\theta }$, describe the products of its elements and the operators commuting with $S_{\theta }+S_{\theta }^{\ast }$.

arXiv commentThis is a preliminary version. Pages 25