Index calculation for Toeplitz plus Hankel operators with piecewise quasi-continuous generating functions

Abstract

An explicit index formula for Toeplitz plus Hankel operators with piecewise quasicontinuous generating functions is obtained. Moreover, for some classes of the operators mentioned conditions of one‐sided invertibility are established.

Some results on the invertibility of Toeplitz plus Hankel operators

“…We proceed with additional preparatory material. The first assertion can be proved analogously to [17], see also [6], and the second one directly follows from the relation (3.4). Remark 4.6.…”

Generalized inverses and solution of equations with Toeplitz plus Hankel operators

“…In the case of piecewise continuous generating functions a and b, Fredholm properties of the operator (1) can be derived by a direct application of results [2,.102], [11,Sections 4.5 and 5.7], [12]. The case of quasi piecewise continuous generating functions has been studied in [14], whereas formulas for the index of the operators (1), considered on different Banach and Hilbert spaces and with various assumptions about the generating functions a and b, have been established in [3,13]. Recently, progress has been made in computation of defect numbers dim ker(T (a) + H(b)) and dim coker (T (a) + H(b)) for various classes of generating functions a and b [1,5].…”

“…Note that this relation has been first used in [1] when studying the dimension of kernels and cokernels of Toeplitz plus Hankel operators with piecewise continuous generating functions a and b. Regardless of [1], the importance of relation (12) for the investigation of Toeplitz plus Hankel operators has been mentioned in [3,Remark 9]. The approach proposed is also applicable to non-homogeneous equations with Wiener-Hopf plus Hankel operators with generating matching functions.…”

Generalized Toeplitz Plus Hankel Operators: Kernel Structure and Defect Numbers

“…As far as the operator T (a) + H(b) is concerned, for a, b ∈ P C its Fredholm properties can be immediately derived by a direct applications of results [4,.102], [13,Sections 4.5 and 5.7], [14]. The case of quasi piecewise continuous generating functions has been studied in [16], whereas formulas for the index of the operators (3) considered on various Banach and Hilbert spaces and with various assumptions about the generating functions a and b have been established in [5,15]. Recently, progress has been made in computation of defect numbers dim ker(T (a) + H(b)) and dim coker (T (a) + H(b)) for various classes of generating functions a and b [3,7].…”