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Monotonicity properties of certain classes of norms

✍ Scribed by Boris Lavric̆

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
761 KB
Volume
259
Category
Article
ISSN
0024-3795

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✦ Synopsis

Let p be a norm on K", where K = R or K = C. If S E K", ' is a nonsingular matrix, let ps be the norm on K", defined by p,(x) = p(Sx) for all r E K". This note gives some conditions on S for which p, has a certain monotonicity property, and characterizes p by monotonicity properties of all norms p, with S in a given group of unitary (orthogonal) matrices. In particular, we obtain the following characterizations: (1) If p is an Zy-norm, 1 < q < ~0, all matrices S are described for which p, is weakly monotonic. (2) If f or each unitary (orthogonal)

matrix S E K" ' the norm p, is quasimonotonic, then p is a positive multiple of the I,-norm.

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