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 $A,B$ be square matrices such that, with respect to the usual operator norm $\|\cdot\|$ ,

$$$$ $\|$ (zI-A)^ ${-1}\|$ = $\|$ (zI-B)^ ${-1}\|$ $\qquad$ (z $\in\CC$ ). $$$$

Then it is known that $1/2\le\|A\|/\|B\|\le 2$ . Are there similar bounds for $\|A^n\|/\|B^n\|$ for $n\ge2$ ? Does the answer change if $A,B$ are diagonalizable? What if $(*)$ holds, not just for the norm $\|\cdot\|$ , 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)
Subjects:	H - N > Linear and multilinear algebra; matrix theory O - Z > Operator theory H - N > Numerical analysis
Research Groups:	Numerical Analysis Group
ID Code:	1119
Deposited By:	Lotti Ekert
Deposited On:	11 May 2011 10:56
Last Modified:	11 May 2011 10:56

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