On the approximation of spectra of linear Hilbert space operators

• Main Author: Hansen, A. C.
• Published: University of Cambridge 2008
• Subjects: 543.5
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.603665

The main topic of this thesis is how to approximate and compute spectra of linear operators on separable Hilbert spaces. We consider several approaches including the finite section method, an infinite-dimensional version of the QR algorithm, as well as pseudospectral techniques. Several new theorems about convergence of the finite section method (and variants of it) for self-adjoint problems are obtained together with a rigorous analysis of the infinite-dimensional QR algorithm for normal operators. To attack the general spectral problem we look to the pseudospectral theory and introduce the complexity index. A generalization of the pseudospectrum is introduced, namely, the <i>n</i>-pseudospectrum. This set behaves very much like the original pseudospectrum, but has the advantage that it approximates the spectrum well for large <i>n. </i>The complexity index is a tool for indicating how complex or difficult it may be to approximate spectra of operators belonging to a certain class. We establish bounds on the complexity indeed and discuss some open problems regarding this new mathematical entity. As the approximation framework also gives rise to several computational methods, we analyze and discuss implementation techniques for algorithms that can be derived from the theoretical model. In particular, we develop algorithms that can compute spectra of arbitrary bounded operators on separable Hilbert spaces, and the exposition is followed by several numerical examples. The thesis also contains a thorough discussion on how to implement the QR algorithm in infinite dimensions. This is supported by numerical computations. These examples reveal several surprisingly nice features of the infinite-dimensional QR algorithm, and this leaves a number of open problems that we debate.