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

G. J. Groenewald, S. ter Horst, J. Jaftha, A. C.M. Ran*

*Corresponding author for this work

## 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 Tω∗ is described. In the case where p= 2 and ω has poles only on the unit circle T, a description is given for when Tω∗ is symmetric and when Tω∗ admits a selfadjoint extension. If in addition ω is proper, it is shown that Tω∗ 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).

Original language English 43 1-23 23 Integral Equations and Operator Theory 91 5 24 Sep 2019 https://doi.org/10.1007/s00020-019-2542-2 Published - Oct 2019