Non-strongly converging approximation methods and the approximation
of pseudospectra
Classically, for a given equation and a sequence of compact
projections which converges strongly to the
identity one studies the sequence of (truncated) equations 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 (called
approximate projection) as a starting point for the definition of
appropriate substitutes which we call -compactness,
-Fredholmness and -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 -pseudospectra can be generalized to the
Banach space case and how this may help to deal with the phenomenon
of jumping pseudospectra.
