A handy formula for the Fredholm index of Toeplitz plus Hankel operators

Abstract

We consider Toeplitz and Hankel operators with piecewise continuous generating functions on l p -spaces and the Banach algebra generated by them. The goal of this paper is to provide a transparent symbol calculus for the Fredholm property and a handy formula for the Fredholm index for operators in this algebra.

Generalized Toeplitz Plus Hankel Operators: Kernel Structure and Defect Numbers

“…The representation (15) can be verified by straightforward computations using relations (9)- (10). The rest of the proof goes through as for Lemma 3.2 of [8].…”

“…6 Fredholmness of generalized Toeplitz plus Hankel operators with piecewise continuous generating functions is presented in [13]. On the other hand, an index formula can be established following ideas of [15]. Moreover, if the generating functions constitute a Fredholm matching pair (a, b), the kernel and cokernel of the operator T (a) + H α (b) can be described using results of Section 4.…”

“…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].…”

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].…”

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

“…As was already mentioned, an index formula for Fredholm Toeplitz plus Hankel operator T (a) + H(b) with generating functions from P C has been obtained only recently [15]. As far as PQC-coefficients are concerned, one can try to reduce the index problem to calculation of the winding number of combinations of harmonic extensions.…”