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

doi:10.1007/s00020-019-2542-2

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

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

This paper contains a further analysis of the Toeplitz-like operators T_ω on H^p 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
Article number	43
Pages (from-to)	1-23
Number of pages	23
Journal	Integral Equations and Operator Theory
Volume	91
Issue number	5
Early online date	24 Sept 2019
DOIs	https://doi.org/10.1007/s00020-019-2542-2
Publication status	Published - Oct 2019

Funding

This work is based on research supported in part by the National Research Foundation of South Africa. Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Funders	Funder number
National Research Foundation
National Research Foundation

Keywords

Adjoint
Symmetric operators
Toeplitz operators
Unbounded operators

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10.1007/s00020-019-2542-2

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title = "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 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{\textquoteright}s, 2013; J Math Anal Appl 440:911–921, 2016).",

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AU - Ran, A. C.M.

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N2 - 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).

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

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