  We examine the common null spaces of families of Herz-Schur multipliers and
apply our results to study jointly harmonic operators and their relation with
jointly harmonic functionals. We show how an annihilation formula obtained in
J. Funct. Anal. 266 (2014), 6473-6500 can be used to give a short proof as well
as a generalisation of a result of Neufang and Runde concerning harmonic
operators with respect to a normalised positive definite function. We compare
the two notions of support of an operator that have been studied in the
literature and show how one can be expressed in terms of the other.
