Functional Analysis and Applications Seminar 

29/06/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Markus Seidel, Chemnitz University of Technology, Germany

Non-strongly converging approximation methods and the approximation of pseudospectra

Classically, for a given equation $Ax=b$ and a sequence of compact projections $𝒫=(P_n)$ which converges strongly to the identity one studies the sequence of (truncated) equations $P_{n}A{P}_n{x}_n=P_{n}b$ in order to find approximate solutions for the initial problem. The theory behind that idea is heavily based on the interactions between compactness, Fredholmness and strong convergence. In the first part of this talk we now turn the table in a sense, and we take a sequence $𝒫=(P_n)$ (called approximate projection) as a starting point for the definition of appropriate substitutes which we call $𝒫$-compactness, $𝒫$-Fredholmness and $𝒫$-strong convergence. On the one hand, this adapted framework permits to develop a theory that mimics the classical one and that provides very similar results on the applicability of the projection method, the stability, and on the asymptotics of norms, condition numbers or pseudospectra. On the other hand, it can be applied to much more general settings since it is detached from the fixed classical notions. The second part picks up the approximation of pseudospectra in more detail. We particular demonstrate how Hansens concept of $(N,\epsilon )$-pseudospectra can be generalized to the Banach space case and how this may help to deal with the phenomenon of jumping pseudospectra.