Discrete convolution operators with discontinuous symbols

Abstract

In [1], the operator ideals analogous to those studied by N. K. Nikol’skii in [2] are introduced and investigated. On the base of the theory of these ideals a presymbol is constructed, the canonical representation is obtained, and the necessary conditions for the Fredholm property are established for operators from a Banach algebra generated by a singular integral Cauchy operator with general discontinuous coefficients in the Lp-space on a circle. In particular, in [1], the operators with arbitrary essentially bounded coefficients are studied. It is well-known that the symbols of discrete convolution operators with summable kernels are continuous on a circle ([3], p. 59). In this paper, we consider the discrete convolution operators, whose kernels belong to a much wider class; the symbols of these operators are arbitrary essentially bounded functions. For Banach algebras of these operators with coefficients stabilizing at infinity in the case p = 2 we obtain the results similar to those from [1].