Finite sections of band operators with slowly oscillating coefficients
The aim of this note is to prove that the finite section method for a band operator \(A\in L(l^2(\mathbb Z))\) with slowly oscillating coefficients \(a_{i,j}\) with respect to the standard basis of \(l^2(\mathbb Z)\) is stable if and only if the operator \(A\) is invertible and if the plus-index of \(A\) is zero. This result generalizes the classical stability criterion for the finite section method for band Toeplitz operators. The finite section method consists in replacing the operator equation \(Au=f\) by the sequences of the linear systems \(R_n AR_n u_n=R_n f,\;n\in \mathbb N\), where \[ R_n :\;l^2(\mathbb Z)\to l^2(\mathbb Z),\;(R_n x)_m= \begin{cases} x_m & \text{if}\;| m| >n, \\ 0 & \text{if}\;| m| >n \end{cases} \] and this method is called stable if the matrices \(R_n AR_n \) are invertible for sufficiently large \(n\) and if the norm of their inverses are uniformly bounded.
scientific article; zbMATH DE number 2116521