Pseudospectra of Linear Operators

Abstract

The following contains mathematical formulae and symbols that may become distorted in ASCII text format. The advent of ever more powerful computers has brought with it a new way of conceiving some of the fundamental eigenvalue problems of applied mathematics. If a matrix or linear operator "A" is far from normal, its eigenvalues or more generally its spectrum may have little to do with its behavor as measured by quantities such as ||A**N|| or ||exp(tA)||. More may be learned by examining the sets in the complex plane known as the "pseudospectra" of A, defined by level curves of the norm of the resolvent, ||(zI - A)**-1||. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly non-normal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.