
Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States
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 Categories:15A60, 15A04, 47A12, 47A30 