### Operator Monotone Functions of Several Variables

- Series
- School of Mathematics Colloquium
- Time
- Thursday, October 18, 2012 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- John McCarthy – Washington University - St. Louis

Self-adjoint $n$-by-$n$ matrices have a natural partial ordering,
namely $ A \leq B $ if the matrix $ B - A$ is positive semi-definite.
In 1934 K. Loewner characterized functions that preserve this ordering;
these functions are called $n$-matrix monotone.
The condition depends on the dimension $n$, but if a function
is $n$-matrix monotone for all $n$, then it must extend analytically
to a function that maps the upper half-plane to itself.
I will describe Loewner's results, and then discuss what happens
if one wants to characterize functions $f$ of two (or more) variables that
are matrix monotone in the following sense:
If $ A = (A_1, A_2)$ and $B = (B_1,B_2)$ are pairs of commuting
self-adjoint
$n$-by-$n$ matrices, with $A_1 \leq B_1 $ and $A_2 \leq B_2$,
then $f(A) \leq f (B)$.
This talk is based on joint work with Jim Agler and Nicholas Young.