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

Abstract

Abstract. This paper is concerned with the applicability of the finite section method to oper-ators belonging to the closed subalgebra of L(L 2 ( R )) generated by operators of multiplicationby piecewise continuous functions in R˙ , convolution operators – also with piecewise continuousgenerating functions – and the flip operator (Ju)(x) = u(−x). For this, a larger algebra ofsequences is introduced, which contains the special sequences we are interested in. There is adirect relationship between the applicability of the finite section method for a given operatorand the invertibility of the corresponding sequence in this algebra. Exploring this relationship,the methods of essentialization, localization and identification of the local algebras throughconstruction of locally equivalent representations are used and so useful invertibility criteriaare derived. Finally, examples are presented, including explicit conditions for the applicabilityof the finite section method to a Wiener-Hopf plus Hankel operator with piecewise continuoussymbols, and some relations between the approximation operators and the limit operator arediscussed.Keywords: Finite section method, Wiener-Hopf operators, Hankel operators.AMS subject classification: 65R20, 47C15.