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

Research output: Contribution to JournalArticleAcademicpeer-review

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 languageEnglish
Article number43
Pages (from-to)1-23
Number of pages23
JournalIntegral Equations and Operator Theory
Volume91
Issue number5
Early online date24 Sep 2019
DOIs
Publication statusPublished - Oct 2019

Fingerprint

Otto Toeplitz
Unit circle
Pole
Toeplitz Operator
Operator
Adjoint Operator
Self-adjoint Extension
Multiplication Operator
Unbounded Operators

Keywords

  • Adjoint
  • Symmetric operators
  • Toeplitz operators
  • Unbounded operators

Cite this

@article{23177ffe6f9e4257a021e1c7a32636d0,
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’s, 2013; J Math Anal Appl 440:911–921, 2016).",
keywords = "Adjoint, Symmetric operators, Toeplitz operators, Unbounded operators",
author = "Groenewald, {G. J.} and {ter Horst}, S. and J. Jaftha and Ran, {A. C.M.}",
year = "2019",
month = "10",
doi = "10.1007/s00020-019-2542-2",
language = "English",
volume = "91",
pages = "1--23",
journal = "Integral Equations and Operator Theory",
issn = "0378-620X",
publisher = "Birkhauser Verlag Basel",
number = "5",

}

A Toeplitz-Like Operator with Rational Symbol Having Poles on the Unit Circle III : The Adjoint. / Groenewald, G. J.; ter Horst, S.; Jaftha, J.; Ran, A. C.M.

In: Integral Equations and Operator Theory, Vol. 91, No. 5, 43, 10.2019, p. 1-23.

Research output: Contribution to JournalArticleAcademicpeer-review

TY - JOUR

T1 - A Toeplitz-Like Operator with Rational Symbol Having Poles on the Unit Circle III

T2 - The Adjoint

AU - Groenewald, G. J.

AU - ter Horst, S.

AU - Jaftha, J.

AU - Ran, A. C.M.

PY - 2019/10

Y1 - 2019/10

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).

AB - 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).

KW - Adjoint

KW - Symmetric operators

KW - Toeplitz operators

KW - Unbounded operators

UR - http://www.scopus.com/inward/record.url?scp=85073223642&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85073223642&partnerID=8YFLogxK

U2 - 10.1007/s00020-019-2542-2

DO - 10.1007/s00020-019-2542-2

M3 - Article

VL - 91

SP - 1

EP - 23

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 5

M1 - 43

ER -