Wiener-Hopf Indices of Unitary-Valued Functions on the Imaginary Axis

A. E. Frazho, A. C.M. Ran*, F. van Schagen

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

This paper is concerned with the Wiener-Hopf indices of unitary-valued rational matrix functions on the imaginary axis. These indices play a role in the Fredholm theory for Wiener-Hopf integral operators. Our main result gives formulas for the Wiener-Hopf indices in terms of the matrices appearing in realizations of the factors in a Douglas-Shapiro-Shields factorization of the unitary-valued function. Two approaches to this problem are presented: one direct approach using operator theoretic methods, and a second approach using the Cayley transform which allows to use results for an analogous problem regarding unitary-valued functions on the unit circle and corresponding Toeplitz operators.

Original languageEnglish
Article number156
Pages (from-to)1-37
Number of pages37
JournalComplex Analysis and Operator Theory
Volume18
Early online date21 Sept 2024
DOIs
Publication statusPublished - 2024

Bibliographical note

Publisher Copyright:
© The Author(s) 2024.

Funding

The work of the second author is supported in part by the National Research Foundation of South Africa (NRF), Grant Number 145688.

FundersFunder number
National Research Foundation145688

    Keywords

    • 47A53
    • 47A56
    • 47A68
    • 47B35
    • Unitary rational matrix functions
    • Wiener-Hopf indices
    • Wiener-Hopf operators

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