Criterion for uniform invertibility of families of strong approximations to one-dimensional singular integral operators with continuous coefficients

Abstract

ABSTRACT. A criterion is obtained for the applicability of the approximation method based on strongly approximating operator families converging to a one-dimensional singular integral operator with coefficients continuous in the circle. Some special cases are considered.KEY WORDS: singular integral operator, compact operator, approximation methods, strong convergence, trigonometrical polynomials.Let F = {t : t E C, [t[ = 1} be the disk in the complex plane C. Let us consider the singular integral operator A = aI + bS + T acting in the space Lv(F) (from now on 1 < p < oo), where a, b E C(F), T is a compact operator in Lv(F), S is the operator of singular integration: / f(r) d,, t r,(s/)(t) = -7 r-t acting in Lv(F), the integral is understood in the principal value sense of Cauchy. For r > 0, r ~ 1, let us define operators with bounded kernels:acting in the space Lv(F ) . The operators Sr converge to S in the strong operator topology as r -o 1. Let us introduce the family of operators Ar = aI + bS,. + T (r > 0, r r 1) acting in the space Lv(P ) . We assume, moreover, that the operator A is invertible. Consider the following problem: To find conditions necessary and sufficient for the operators A~ to be invertible for all values of r sufficiently close to 1, and for the strong convergence of the corresponding operators A~ "1 to A -1 as r -* 1. In this case we say that the approximation method over the family of operators A~ is applicable to the operator A as r -* 1. This problem is similar to the convergence problem for the projection method [1, p. 90].In the present paper we study more general strong approximations to the operator of singular integration. In contrast to the case described above, sequences of approximating operators are considered. This assl~mption is only made to simplify the assertions. The generalization to the case of families parametrized by a continuous parameter does not encounter any difficulties. The solution of the problem posed above is given as an application of the general result obtained here.Other approximations to the operator of singular integration were considered in our papers [2][3][4].w Let A, An (n = 1,2, ...) be linear continuous operators acting in some Banach space. The notation s-lira A. = A means that the operators An converge to A as n --* oo in the strong operator topology. Besides, if such a relationship is true for the adjoint operators, we write s*-hm An = A.