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→trf(A[I]) is supermodular, meaning that trf(A[I])+trf(A[J]) ≤trf(A[I∪J])+trf(A[I∩J]), where A[I] denotes the I × I principal submatrix of A. We discuss extensions to self-adjoint operators on infinite dimensional Hilbert space and to M-matrices. We also discuss an application to CUR approximation of nonnegative hermitian matrices.