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W. Auzinger:
"Matrix exponentials and normalized numerical range";
Talk: Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck (invited); 2004-03-26.

For a matrix A in C^(nxn), the 2-norm of the matrix exponential exp(tA) is bounded by K=1 for all t>0 if and only if the numerical range R(A) is contained in the left complex half plane (dissipativity).

We discuss a nontrivial generalization of this basic statement. In particular, we introduce a 'normalized version' R_N(A) of the numerical range, which is a subset of the complex unit circle. We show that the set R_N(A) is useful in the characterization of matrices with uniformly bounded exponentials, i.e., ||exp(tA)||_2 <= K with K>1. These results are related to the Kreiss matrix theorem.

Furthermore we consider a class of sectorial matrices which typically arise after semi-discretization of parabolic PDEs. The standard case, where the numerical range R(A) is contained in a sector S of the left half-plane, is equivalent to a certain norm estimate for the resolvent (zI-A)^{-1} for z not in S. The validity of the same estimate, but with a factor K>1, is shown to be equivalent to an inclusion for the normalized numerical range R_N(A).

These criteria in terms of the location of R_N(A) can be interpreted as certain strengthened Cauchy-Schwarz inequalities for arbitrary pairs u,Au (u in C^n).

http://publik.tuwien.ac.at/files/pub-tm_1779.pdf