On residualities in the set of Markov operators on 𝒞₁

Abstract

We show that the set of those Markov operators on the Schatten class C 1 \mathcal {C}_1 such that lim n → ∞ ‖ P n − Q ‖ = 0 \lim _{n \to \infty } \| P^n - Q \| = 0 , where Q Q is one-dimensional projection, is norm open and dense. If we require that the limit projections must be on strictly positive states, then such operators P P form a norm dense G δ G_{\delta } . Surprisingly, for the strong operator topology operators the situation is quite the opposite.