Quasidiagonality and the finite section method
Nathanial P. Brown
TL;DR
The paper develops a comprehensive framework showing that the finite section method robustly approximates quasidiagonal operators on Hilbert spaces, with precise existence, uniqueness, and convergence results. By exploiting asymptotic commutativity, C$^*$-algebraic techniques, and spectral theory, it provides stable and regularizable finite-section sequences, uniform convergence of pseudospectra and singular values, and explicit convergence rates in key cases. Special attention is given to normal and self-adjoint operators, as well as exact C$^*$-algebras, yielding stronger spectral-convergence results and practical finite-dimensional decompositions. The final sections tailor the theory to unilateral and bilateral band operators, offering concrete algorithms (including Berg's technique) and computable error bounds, along with a new proof that the spectra of quasidiagonal weighted shifts attain the maximal possible extent. Collectively, the results enable reliable numerical analysis and spectrum computation for a wide class of quasidiagonal operators, including concrete strategies for band-dominated systems.
Abstract
Quasidiagonal 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 quasidiagonality 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.