A Stability Problem Involving Approximate Identities, Discrete Convolution Operators, Singular Integral Operators, and Finite Sections

Let $n \in \mathbb{N}$ tend towards infinity and $r \in [0,1)$ tend towards 1 with the condition that $n(1-r) \rightarrow \lambda$ for some fixed $\lambda \in (0,\infty).$ A sequence $(F_{n,r})$ of bounded linear operators on a Hilbert space is called $\lambda-$stable if for all sufficiently large $n$ and all $r$ sufficiently close to 1 such that $n(1-r)$ is sufficiently close to $\lambda$, each $F_{n,r}$ is invertible and these inverses are uniformly bounded. We consider the $\lambda-$stability problem for sequences arising from a $C^*-$algebra containing discrete convolution operators, singular integral operators, and their finite sections. Our main result is that a sequence in a certain $C^*-$ algebra is $\lambda-$stable if and only if a certain collection of operators given by strong limits is invertible. As an application, we relate this result to approximate identities and discuss several concrete examples such as finite sections of Toeplitz operators $(T_n(k_\omega a))$ whose symbols are approximate identities applied to piecewise continuous functions and finite sections of singular integral operators