The Mathematical Institute, University of Oxford, Eprints Archive

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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