Abstract
This paper contains a further analysis of the Toeplitz-like operators T_omega on H^p with rational symbol omega 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 ^* is described. In the case where p=2 and omega has poles only on the unit circle {mathbb {T}}, a description is given for when T_omega ^* is symmetric and when T_omega ^* admits a selfadjoint extension. If in addition omega is proper, it is shown that T_omega ^* 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).
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