TY - JOUR

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

T2 - Fredholm Properties

AU - Groenewald, G. J.

AU - ter Horst, S.

AU - Jaftha, J.

AU - Ran, A. C.M.

PY - 2020/11/13

Y1 - 2020/11/13

N2 - This paper concerns the analysis of an unbounded Toeplitz-like operator generated by a rational matrix function having poles on the unit circle T. It extends the analysis of such operators generated by scalar rational functions with poles on T found in Groenewald et al. (Oper Theory Adv Appl 271:239–268, 2018; Oper Theory Adv Appl 272:133–154, 2019; Integr Equ Oper Theory 91, 2019). A Wiener–Hopf type factorization of rational matrix functions with poles and zeroes on T is proved and then used to analyze the Fredholm properties of such Toeplitz-like operators. A formula for the index, based on the factorization, is given. Furthermore, it is shown that the determinant of the matrix function having no zeroes on T is not sufficient for the Toeplitz-like operator to be Fredholm, in contrast to the classical case.

AB - This paper concerns the analysis of an unbounded Toeplitz-like operator generated by a rational matrix function having poles on the unit circle T. It extends the analysis of such operators generated by scalar rational functions with poles on T found in Groenewald et al. (Oper Theory Adv Appl 271:239–268, 2018; Oper Theory Adv Appl 272:133–154, 2019; Integr Equ Oper Theory 91, 2019). A Wiener–Hopf type factorization of rational matrix functions with poles and zeroes on T is proved and then used to analyze the Fredholm properties of such Toeplitz-like operators. A formula for the index, based on the factorization, is given. Furthermore, it is shown that the determinant of the matrix function having no zeroes on T is not sufficient for the Toeplitz-like operator to be Fredholm, in contrast to the classical case.

KW - Fredholm properties

KW - Rational matrix functions

KW - Toeplitz operators

KW - Unbounded operators

KW - Wiener–Hopf factorization

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U2 - 10.1007/s11785-020-01040-z

DO - 10.1007/s11785-020-01040-z

M3 - Article

AN - SCOPUS:85095950587

VL - 15

SP - 1

EP - 29

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

SN - 1661-8254

IS - 1

M1 - 1

ER -