Monotone Matrix functions in several variables

In 1934, K. Loewner characterized functions that preserve

matrix inequalities,

i.e.\ those f with the property that whenever A and B are self-adjoint

matrices of the same dimension,

with $A \leq B$, then $f(A) \leq f(B)$.

In this talk, I shall discuss how to characterize monotone matrix

functions of several variables,

namely functions f with the property that if $A = (A_1, \dots , A_n) $

is an n-tuple of commuting self-adjoint matrices,

and $B = (B_1, \dots, B_n)$ is another, with each $A_i \leq B_i$, then

$f(A) \leq f(B)$.