Pseudospectra and norm bounds

Student thesis: Master thesis (including HD thesis)

  • Lars V. Iversen
  • Dan V. Jensen
  • Ove L. Sandau
4. term, Mathematics, Master (Master Programme)
Let A be a bounded operator in a Hilbert space. It is of interest to give estimates for ||f(A)||, in particular when f is a polynomial. The objective of this report is to give such bounds for ||f(A)|| based on pseudospectra and the Kreiss matrix theorem, of which the latter is generalized to bounded operators with spectrum \sigma(A) in arbitrary compact sets \sigma(A) \subseteq \Omega \subset C, and to address the question of computation of quantities related to norm bounds, particularly related to pseudospectra. To associate an operator with a continuous function, the Dunford calculus is used, and complex analysis and in particular the Cauchy integral formula will play an important role. In the first chapter we will present some classical results from complex analysis, e.g. the maximum modulus principle, the open mapping principle and the Riemann mapping theorem. In the second chapter we associate pseudospectra and the Kreiss constant, and through the conformal mapping from the Riemann mapping theorem another definition of the Kreiss constant is presented, and it is shown that the two definitions are equivalent in a sense. The Faber polynomials, too, are related to the Kreiss constant in the sense that they, too, are defined through the conformal mapping from the Riemann mapping theorem. These are defined in the third chapter, and generalizations of the Kreiss matrix theorem to Faber polynomials and polynomials in general are proven. Pseudospectra are particularly useful for non-normal operators. In chapter four a discretization of the differential operator by Chebyshev differential methods is considered, and implementation and some properties of the non-normal Chebyshev differentiation matrix are presented. The last chapter considers computation of the pseudospectral abscissa, which e.g. yields a lower bound for the transient behavior of ||exp(tA)||. The criss-cross algorithm is presented, and some properties concerning convergence are proven.
Publication date2009
Number of pages112
Publishing institutionInstitut for Matematiske Fag
ID: 17652479