A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties

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

doi:10.1007/978-3-030-04269-1_10

A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties

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

^*Corresponding author for this work

Mathematics

Research output: Chapter in Book / Report / Conference proceeding › Chapter › Academic › peer-review

85 Downloads (Pure)

Abstract

In this paper a definition is given for an unbounded Toeplitzlike operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.

Original language	English
Title of host publication	Operator Theory, Analysis and the State Space Approach
Subtitle of host publication	In Honor of Rien Kaashoek
Editors	Harm Bart, Sanne ter Horst, André C.M. Ran, Hugo J. Woerdeman
Publisher	Birkhäuser
Pages	239-268
Number of pages	30
ISBN (Electronic)	9783030042691
ISBN (Print)	9783030042684
DOIs	https://doi.org/10.1007/978-3-030-04269-1_10
Publication status	Published - 2018

Publication series

Name	Operator Theory: Advances and Applications
Publisher	Birkhäuser
Volume	271
ISSN (Print)	0255-0156
ISSN (Electronic)	2296-4878

Funding

This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 90670 and 93406). Part of the research was done during a sabbatical of the third author, in which time several research visits to VU Amsterdam and North-West University were made. Support from University of Cape Town and the Department of Mathematics, VU Amsterdam is gratefully acknowledged. Acknowledgement. The present 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. The present 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. We would also like to thank the anonymous referee for the useful suggestions, corrections and comments.

Funders	Funder number
National Research Foundation
National Research Foundation of South Africa	90670, 93406
National Research Foundation

Keywords

Fredholm properties
Toeplitz operators
Unbounded operators

UN SDGs

This output contributes to the following UN Sustainable Development Goals (SDGs)

Access to Document

10.1007/978-3-030-04269-1_10

Groenewald2018_Chapter_AToeplitz-likeOperatorWithRationalSymbolFinal published version, 566 KBLicence: Article 25fa Dutch Copyright Act

Persistent URL (handle)

Persistent link

Cite this

Groenewald, G. J., ter Horst, S., Jaftha, J., & Ran, A. C. M. (2018). A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties. In H. Bart, S. ter Horst, A. C. M. Ran, & H. J. Woerdeman (Eds.), Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek (pp. 239-268). (Operator Theory: Advances and Applications; Vol. 271). Birkhäuser. https://doi.org/10.1007/978-3-030-04269-1_10

Groenewald, G. J. ; ter Horst, S. ; Jaftha, J. et al. / A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties. Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek. editor / Harm Bart ; Sanne ter Horst ; André C.M. Ran ; Hugo J. Woerdeman. Birkhäuser, 2018. pp. 239-268 (Operator Theory: Advances and Applications).

@inbook{ea6009a7d7a24ed18b40c2c1982a1eb1,

title = "A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties",

abstract = "In this paper a definition is given for an unbounded Toeplitzlike operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.",

keywords = "Fredholm properties, Toeplitz operators, Unbounded operators",

author = "Groenewald, \{G. J.\} and \{ter Horst\}, S. and J. Jaftha and Ran, \{A. C.M.\}",

year = "2018",

doi = "10.1007/978-3-030-04269-1\_10",

language = "English",

isbn = "9783030042684",

series = "Operator Theory: Advances and Applications",

publisher = "Birkh{\"a}user",

pages = "239--268",

editor = "Harm Bart and \{ter Horst\}, Sanne and Ran, \{Andr{\'e} C.M.\} and Woerdeman, \{Hugo J.\}",

booktitle = "Operator Theory, Analysis and the State Space Approach",

}

Groenewald, GJ, ter Horst, S, Jaftha, J & Ran, ACM 2018, A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties. in H Bart, S ter Horst, ACM Ran & HJ Woerdeman (eds), Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek. Operator Theory: Advances and Applications, vol. 271, Birkhäuser, pp. 239-268. https://doi.org/10.1007/978-3-030-04269-1_10

A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties. / Groenewald, G. J.; ter Horst, S.; Jaftha, J. et al.
Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek. ed. / Harm Bart; Sanne ter Horst; André C.M. Ran; Hugo J. Woerdeman. Birkhäuser, 2018. p. 239-268 (Operator Theory: Advances and Applications; Vol. 271).

Research output: Chapter in Book / Report / Conference proceeding › Chapter › Academic › peer-review

TY - CHAP

T1 - A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties

AU - Groenewald, G. J.

AU - ter Horst, S.

AU - Jaftha, J.

AU - Ran, A. C.M.

PY - 2018

Y1 - 2018

N2 - In this paper a definition is given for an unbounded Toeplitzlike operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.

AB - In this paper a definition is given for an unbounded Toeplitzlike operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.

KW - Fredholm properties

KW - Toeplitz operators

KW - Unbounded operators

UR - https://www.scopus.com/pages/publications/85060173385

UR - https://www.scopus.com/inward/citedby.url?scp=85060173385&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-04269-1_10

DO - 10.1007/978-3-030-04269-1_10

M3 - Chapter

AN - SCOPUS:85060173385

SN - 9783030042684

T3 - Operator Theory: Advances and Applications

SP - 239

EP - 268

BT - Operator Theory, Analysis and the State Space Approach

A2 - Bart, Harm

A2 - ter Horst, Sanne

A2 - Ran, André C.M.

A2 - Woerdeman, Hugo J.

PB - Birkhäuser

ER -

Groenewald GJ, ter Horst S, Jaftha J, Ran ACM. A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties. In Bart H, ter Horst S, Ran ACM, Woerdeman HJ, editors, Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek. Birkhäuser. 2018. p. 239-268. (Operator Theory: Advances and Applications). Epub 2018 Dec 31. doi: 10.1007/978-3-030-04269-1_10

A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties

Abstract

Publication series

Funding

Keywords

UN SDGs

Access to Document

Persistent URL (handle)

Other files and links

Fingerprint

Cite this