Some results on the invertibility of Toeplitz plus Hankel operators

Abstract

Abstract. The paper deals with the invertibility of Toeplitz plus Hankel operators T (a)+H(b) acting on classical Hardy spaces on the unit circle T. It is supposed that the generating functions a and b satisfy the condition a(t)a(1/t) = b(t)b(1/t), t ∈ T. Special attention is paid to the case of piecewise continuous generating functions. In some cases the dimensions of null spaces of the operator T (a) + H(b) and its adjoint are described.

Kernels of Wiener–Hopf plus Hankel Operators with Matching Generating Functions

“…In Section 2 we present a decomposition of the kernel of W (g) with a generating matching function g. These results are used in Section 3 in order to derive an efficient description of the kernels ker(W (a) ± H(b)), p ∈ [1, ∞] and the cokernels coker (W (a) ± H(b)), p ∈ [1, ∞). Similar results for Toeplitz plus Hankel operators have been obtained in [9,10], and generalized Toeplitz plus Hankel operators are considered in [11]. However, all the relevant operators in [9,10,11] are Fredholm.…”

Generalized inverses and solution of equations with Toeplitz plus Hankel operators

“…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]. The more delicate problem of the description of the spaces ker(T (a) + H(b)) and coker (T (a) + H(b)) has been considered [5,6].…”

“…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]. The more delicate problem of the description of the spaces ker(T (a) + H(b)) and coker (T (a) + H(b)) has been considered [5,6]. In particular, for generating functions a and b, satisfying the so-called matching condition (see condition (12) [7,8,10] but, as a rule, only some approximation methods of their solutions have been systematically studied so far.…”

Generalized Toeplitz Plus Hankel Operators: Kernel Structure and Defect Numbers

“…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]. Moreover, a more delicate problem of the description of the spaces ker(T (a) + H(b)) and coker (T (a) + H(b)) has been considered [7,8].…”