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 - 2021/2
Y1 - 2021/2
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
SN - 1661-8254
VL - 15
SP - 1
EP - 29
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
IS - 1
ER -