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Journal of the London Mathematical Society Advance Access originally published online on March 12, 2009
Journal of the London Mathematical Society 2009 79(2):511-528; doi:10.1112/jlms/jdn085
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© 2009 London Mathematical Society

Super-identical pseudospectra

Maxime Fortier Bourque and Thomas Ransford

Département de mathématiques et de statistique
Université Laval
Québec (QC)
Canada G1V 0A6
maxime.fortier-bourque.1@ulaval.ca

The complex N x N matrices A and B are said to have super-identical pseudospectra if, for each z isin C, the singular values of A – zI are the same as those of B – zI. We explore this condition and its consequences. On the positive side, drawing on ideas from invariant theory, we prove that there exists an integer m = m(N) such that ‘almost every’ m-tuple of N x N matrices with super-identical pseudospectra contains a pair that are unitarily equivalent. On the negative side, we present an example of a pair of non-derogatory 4 x 4 matrices A and B with super-identical pseudospectra such that ||A2|| != ||B2||.

2000 Mathematics Subject Classification 15A18 (primary), 16R30 (secondary).

The first author was supported by an NSERC undergraduate student research award. The second author was supported by grants from NSERC (Canada), FQRNT (Québec), and the Canada research chairs program.

Received August 22, 2008; published online March 12, 2009.


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