Quasi-diagonality and the finite section method

Quasi-diagonal operators on a Hilbert space are a large and important class (containing all self-adjoint operators for instance). They are also perfectly suited for study via the finite section method (a particular Galerkin method). Indeed, the very definition of quasi-diagonality yields finite sections with good convergence properties. Moreover, simple operator theory techniques yield estimates on certain rates of convergence. In the case of quasidiagonal band operators both the finite sections and rates of convergence are explicitly given.