### 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 Sep 2019 |

DOIs | |

Publication status | Published - Oct 2019 |

### Fingerprint

### Keywords

- Adjoint
- Symmetric operators
- Toeplitz operators
- Unbounded operators

### Cite this

*Integral Equations and Operator Theory*,

*91*(5), 1-23. [43]. https://doi.org/10.1007/s00020-019-2542-2

}

*Integral Equations and Operator Theory*, vol. 91, no. 5, 43, pp. 1-23. https://doi.org/10.1007/s00020-019-2542-2

**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.

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