A Toeplitz-like operator with rational symbol having poles on the unit circle. III. The adjoint

Abstract

This paper contains a further analysis of the Toeplitz-like operators Tω on Hp with rational symbol 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∗ω is described. In the case where p=2 and ω has poles only on the unit circle T , a description is given for when T∗ω is symmetric and when T∗ω admits a selfadjoint extension. If in addition ω is proper, it is shown that 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)