Finite Section Method in some Algebras of Multiplication and Convolution Operators and a Flip

Abstract

This paper is concerned with the applicability of the finite section method to operators belonging to the closed subalgebra of $\mathcal{L}(L^2(\mathbb{R}))$ generated by operators of multiplication by piecewise continuous functions in $\mathbb{R}$, convolution operators - also with piecewise continuous generating functions - and the flip operator $(Ju)(x)=u(–x)$. For this, a larger algebra of sequences is introduced, which contains the special sequences we are interested in. There is a direct relationship between the applicability of the finite section method for a given operator and the invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras through construction of locally equivalent representations are used and so useful invertibility criteria are derived. Finally, examples are presented, including explicit conditions for the applicability of the finite section method to a Wiener-Hopf plus Hankel operator with piecewise continuous symbols, and some relations between the approximation operators and the limit operator are discussed.