Type: Article
Publication Date: 2004-02-01
Citations: 6
DOI: https://doi.org/10.4153/cjm-2004-007-4
Abstract Every norm v on C n induces two norm numerical ranges on the algebra M n of all n × n complex matrices, the spatial numerical range where v D is the norm dual to v , and the algebra numerical range where is the set of states on the normed algebra M n under the operator norm induced by v . For a symmetric norm v , we identify all linear maps on M n that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, i.e. , linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if v is not the ℓ 1 , ℓ 2 , or ℓ ∞ norms, then the linear maps that preserve either numerical range or either set of states are “inner”, i.e. , of the form A ⟼ Q * AQ , where Q is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the ℓ 1 and the ℓ ∞ norms, the results are quite different.