Generalized inverses and solution of equations with Toeplitz plus Hankel operators

Abstract

Considered is the equationwhere T (a) and H(b), a, b ∈ L ∞ (T) are, respectively, Toeplitz and Hankel operators acting on the classical Hardy spaces H p (T), 1 < p < ∞. If the generating functions a and b satisfy the so-called matching condition [1,2],an efficient method for solving equation (⋆) is proposed. The method is based on the Wiener-Hopf factorization of the scalar functions c(t) = a(t)b −1 (t) and d(t) = a(t)b −1 (1/t) and allows one to find all solutions of the equations mentioned.

Wiener-Hopf plus Hankel operators: Invertibility Problems

“…The aim of this work is to find conditions for onesided invertibility, invertibility and generalized invertibility of the operators W(a, b) and to provide efficient representations for the corresponding inverses when generating functions a and b satisfy the matching condition (5). Similar problems for Toeplitz plus Hankel operators have been recently discussed in [1,2,8,9,10,11]. However, the situation with Wiener-Hopf plus Hankel operators has some special features.…”

“…An operator A is called generalized invertible if there exists an operator A −1 g referred to as generalized inverse for A, such that In this section we construct a generalized inverse for operator W (a) + H(b) provided that the operator B is generalized invertible and a generalized inverse of B can be represented in a special form. The following theorem has been proved in [10] in the case of Toeplitz plus Hankel operators. For Wiener-Hopf plus Hankel operators the proof literally repeats all constructions there and is omitted here.…”

Invertibility Issues for Toeplitz plus Hankel Operators and Their Close Relatives

“…The representation (3.7) has been used in [16], to derive the description of the kernels of the operators T (a) ± H(b). It can be also employed to study one-sided or generalized invertibility of Toeplitz plus Hankel operators and to construct the corresponding one-sided and generalized inverses [17,19,23].…”

“…VD & BS[17]) Assume that (a, b) is a matching pair with the subordinated pair (c, d) and B is generalized invertible operator, which has a generalized inverse B −1 of the form…”

“…[17]:(a) If κ 1 ≥ 0 and κ 2 ≥ 0, then B has generalized inverse of the form (4 1). withA = T −1 r (c)T ( a −1 )T −1 r (d), B = −T −1 r(c) and D = T −1 r (d).…”