## 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.

Language | Danish |
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Publication date | 2009 |

Number of pages | 112 |

Publishing institution | Institut for Matematiske Fag |