  We extend the multiplicative submodularity of the principal determinants of a
nonnegative definite hermitian matrix to other spectral functions. We show that
if $f$ is the primitive of a function that is operator monotone on an interval
containing the spectrum of a hermitian matrix $A$, then the function $I\mapsto
{\rm tr} f(A[I])$ is supermodular, meaning that ${\rm tr} f(A[I])+{\rm tr}
f(A[J])\leq {\rm tr} f(A[I\cup J])+{\rm tr} f(A[I\cap J])$, where $A[I]$
denotes the $I\times I$ principal submatrix of $A$. We discuss extensions to
self-adjoint operators on infinite dimensional Hilbert space and to
$M$-matrices. We discuss an application to CUR approximation of nonnegative
hermitian matrices.
