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)



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 09:56
Last Modified:29 May 2015 18:48

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