  This article provides sufficient conditions for positive maps on the Schatten
classes $\mathcal J_{p}, 1\le p<\infty$ of bounded operators on a separable
Hilbert space such that a corresponding Perron-Frobenius theorem holds. With
applications in quantum information theory in mind sufficient conditions are
given for a trace preserving, positive map on $\mathcal J_{1}$, the space of
trace class operators, to have a unique, strictly positive density matrix which
is left invariant under the map. Conversely to any given strictly positive
density matrix there are trace preserving, positive maps for which the density
matrix is the unique Perron-Frobenius vector.
