Finite section approximations in an algebra of convolution, multiplication and flip operators on [formula omitted

Abstract

This paper is concerned with the approximation of solutions of operator equations using the finite sections method with the operators belonging to the closed subalgebra of B(Lp(R)), 1<p<∞, generated by operators of multiplication by piecewise continuous functions in Ṙ, operators of convolution by piecewise continuous Fourier multipliers and the flip operator. This algebra includes Wiener–Hopf and Hankel operators with piecewise continuous symbols. To prove the result, we use algebraic techniques and introduce a larger algebra of sequences, which contains the special sequences we are interested and the usual operator algebra generated by the operators of multiplication, convolution and flip. There is a direct relationship between the applicability of the finite section method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship and using local principles, we construct locally equivalent representations that allow to derive invertibility criteria.