Finite rank harmonic operator-valued functions

The purpose of this paper is to characterize when a harmonic function with values in the finite rank operators on a Hilbert space is expressible as a harmonic matrix-valued function. We show that harmonic function with values in the rank 1 normal operators is expressible as a harmonic matrix-valued function. We also prove that for any natural number, n, a harmonic function with values in the rank n non-negative operators is expressible as a matrix-valued function and we give examples showing that these decomposition theorems fail when various hypotheses are relaxed.