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

## TL;DR

This paper establishes necessary and sufficient conditions for the stability of the finite section method applied to a class of singular integral operators with operator-valued coefficients and a flip, using $C^*$-algebra techniques.

## Contribution

It generalizes previous stability results by including flip operators and operator-valued coefficients in the analysis of singular integral operators.

## Key findings

- Provides a $C^*$-algebra based stability criterion.
- Characterizes stability via invertibility of associated operators.
- Extends classical results to operators with flip and PQC-coefficients.

## Abstract

We establish necessary and sufficient conditions for the stability of the finite section method for operators belonging to a certain $C^*$-algebra of operators acting on the Hilbert space $l^2_H(\mathbb{Z})$ of $H$-valued sequences where $H$ is a given Hilbert space. Identifying $l^2_H(\mathbb{Z})$ with the $L^2_H$-space over the unit circle, the $C^*$-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 $C^*$-algebra methods and says that a sequence of finite sections is stable if and only if certain operators associated with that sequence (via $^*$-homomorphisms) are invertible.