Harnackʹs theorem for harmonic compact operator-valued functions

In this paper we show that harmonic compact operator-valued functions are characterized by having harmonic diagonal matrix coefficients in any choice of basis. We also give an example which shows that an operator-valued function with values outside the compact operators can have harmonic diagonal matrix coefficients in any choice of basis without being a harmonic operator-valued function. We use our harmonic matrix coefficients characterization to establish a Harnackʹs theorem for an increasing sequence of harmonic compact self-adjoint operator-valued functions and we show that this Harnackʹs theorem need not hold when the compactness restriction is dropped.