A Toeplitz-Like Operator with Rational Symbol Having Poles on the Unit Circle III: The Adjoint

Abstract

Abstract This paper contains a further analysis of the Toeplitz-like operators $$T_\omega $$ T ω on $$H^p$$ H p with rational symbol $$\omega $$ ω having poles on the unit circle that were previously studied in Groenewald (Oper Theory Adv Appl 271:239–268, 2018; Oper Theory Adv Appl 272:133–154, 2019). Here the adjoint operator $$T_\omega ^*$$ T ω ∗ is described. In the case where $$p=2$$ p = 2 and $$\omega $$ ω has poles only on the unit circle $${\mathbb {T}}$$ T , a description is given for when $$T_\omega ^*$$ T ω ∗ is symmetric and when $$T_\omega ^*$$ T ω ∗ admits a selfadjoint extension. If in addition $$\omega $$ ω is proper, it is shown that $$T_\omega ^*$$ T ω ∗ coincides with the unbounded Toeplitz operator defined by Sarason (Integr Equ Oper Theory 61:281–298, 2008) and studied further by Rosenfeld (Classes of densely defined multiplication and Toeplitz operators with applications to extensions of RKHS’s, 2013; J Math Anal Appl 440:911–921, 2016).