  Let ${\Bbb G}$ be a locally compact quantum group and ${\mathcal T}(L^2({\Bbb
G}))$ be the Banach algebra of trace class operators on $L^2({\Bbb G})$ with
the convolution induced by the right fundamental unitary of ${\Bbb G}$. We
study the space of harmonic operators $\widetilde{\mathcal H}_\omega$ in
${\mathcal B}(L^2({\Bbb G}))$ associated to a contractive element $\omega\in
{\mathcal T}(L^2({\Bbb G}))$. We characterize the existence of non-zero
harmonic operators in ${\mathcal K}(L^2({\Bbb G}))$ and relate them with some
properties of the quantum group ${\Bbb G}$, such as finiteness, amenability and
co-amenability.
