Submodular spectral functions of principal submatrices of a hermitian matrix, extensions and applications

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 $I\mapsto {\rm tr} f(A[I])$issupermodular,meaningthat${\rm tr} f(A[I])+{\rm tr} f(A[J])\leq {\rm tr} f(A[I\cup J])+{\rm tr} f(A[I\cap J])$,where$A[I]$denotesthe$I\times I$principalsubmatrixof$A$. 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.