Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States

Abstract

AbstractEvery norm v on Cn induces two norm numerical ranges on the algebra Mn of all n × n complex matrices, the spatial numerical rangewhere vD is the norm dual to v, and the algebra numerical rangewhere is the set of states on the normed algebra Mn under the operator norm induced by v. For a symmetric norm v, we identify all linear maps on Mn 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 ℓ1, ℓ2, or ℓ∞ norms, then the linear maps that preserve either numerical range or either set of states are “inner”, i.e., of the formA ⟼ 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 ℓ1 and the ℓ∞ norms, the results are quite different.