Finite Section Method for singular integrals with operator-valued PQC-coefficients and a flip

Abstract

We establish necessary and sufficient conditions for the stability of the finite section method for operators belonging to a certain $C^{\ast }$-algebra of operators acting on the Hilbert space $l_H^2(\mathbb{Z})$ of $H$-valued sequences where $H$ is a given Hilbert space. Identifying $l_H^2(\mathbb{Z})$ with the $L_H^2$-space over the unit circle, the $C^{\ast }$-algebra in question is the one which contains all singular integral operators with flip and piecewise quasicontinous $\mathcal{L}(H)$-valued generating functions on the unit circle. The result is a generalization of an older result where the same problem, but without the flip operator was considered. The stability criterion is obtained via $C^{\ast }$-algebra methods and says that a sequence of finite sections is stable if and only if certain operators associated with that sequence (via ${}^{\ast }$-homomorphisms) are invertible.

References in corpus (1)

• Factorization of a class of Toeplitz + hankel operators and the A_p-condition