On Pseudospectra and Power Growth
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...
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| Format: | Report |
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Unspecified
2006
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| _version_ | 1826279908723130368 |
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| author | Ransford, T |
| author_facet | Ransford, T |
| author_sort | Ransford, T |
| collection | OXFORD |
| description | 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 |
| first_indexed | 2024-03-07T00:05:50Z |
| format | Report |
| id | oxford-uuid:7781c019-d4b8-45bf-b3c7-0340a51c28f7 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:05:50Z |
| publishDate | 2006 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:7781c019-d4b8-45bf-b3c7-0340a51c28f72022-03-26T20:24:29ZOn Pseudospectra and Power GrowthReporthttp://purl.org/coar/resource_type/c_93fcuuid:7781c019-d4b8-45bf-b3c7-0340a51c28f7Mathematical Institute - ePrintsUnspecified2006Ransford, TThe 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 |
| spellingShingle | Ransford, T On Pseudospectra and Power Growth |
| title | On Pseudospectra and Power Growth |
| title_full | On Pseudospectra and Power Growth |
| title_fullStr | On Pseudospectra and Power Growth |
| title_full_unstemmed | On Pseudospectra and Power Growth |
| title_short | On Pseudospectra and Power Growth |
| title_sort | on pseudospectra and power growth |
| work_keys_str_mv | AT ransfordt onpseudospectraandpowergrowth |