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

Published:2004-02-01
Printed: Feb 2004
• Chi-Kwong Li
• Ahmed Ramzi Sourour
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## Abstract

Every norm $\nu$ on $\mathbf{C}^n$ induces two norm numerical ranges on the algebra $M_n$ of all $n\times n$ complex matrices, the spatial numerical range $$W(A)= \{x^*Ay : x, y \in \mathbf{C}^n,\nu^D(x) = \nu(y) = x^*y = 1\},$$ where $\nu^D$ is the norm dual to $\nu$, and the algebra numerical range $$V(A) = \{ f(A) : f \in \mathcal{S} \},$$ where $\mathcal{S}$ is the set of states on the normed algebra $M_n$ under the operator norm induced by $\nu$. For a symmetric norm $\nu$, 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, {\it i.e.}, linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\nu$ is not the $\ell_1$, $\ell_2$, or $\ell_\infty$ norms, then the linear maps that preserve either numerical range or either set of states are inner'', {\it i.e.}, of the form $A\mapsto Q^*AQ$, 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 $\ell_1$ and the $\ell_\infty$ norms, the results are quite different.
 Keywords: Numerical range, numerical radius, state, isometry
 MSC Classifications: 15A60 - Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 15A04 - Linear transformations, semilinear transformations 47A12 - Numerical range, numerical radius 47A30 - Norms (inequalities, more than one norm, etc.)