# On Pseudospectra and Power Growth

Ransford, Thomas (2006) On Pseudospectra and Power Growth. Technical Report. Unspecified. (Submitted)

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## Abstract

The celebrated Kreiss matrix theorem is one of several results relating the norms of the powers of a matrix to its pseudospectra (i.e. the level curves of the norm of the resolvent). But to what extent do the pseudospectra actually determine the norms of the powers? Specifically, let be square matrices such that, with respect to the usual operator norm ,

(zI-A)^=(zI-B)^ (z).

Then it is known that . Are there similar bounds for for ? Does the answer change if are diagonalizable? What if holds, not just for the norm , but also for higher-order singular values? What if we use norms other than the usual operator norm? The answers to all these questions turn out to be negative, and in a rather strong sense.

The research was supported by grants from NSERC and the Canada Research Chairs program

Item Type: Technical Report (Technical Report) H - N > Linear and multilinear algebra; matrix theoryO - Z > Operator theoryH - N > Numerical analysis Numerical Analysis Group 1119 Lotti Ekert 11 May 2011 09:56 29 May 2015 18:48

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