On the norm continuity of transition semigroups in Hilbert spaces

Abstract

The solution semigroup for certain anisotropic heat equations on an infinite dimensional Hilbert space can be defined, e.g., by a limit of finite dimensional Gaussian semigroups. Unlike the heat equation in a finite dimensional Euclidean space, the solution semigroup is known not to be differentiable (and, a fortiori, not analytic). The present paper improves this result and shows that the semigroup is in fact not norm continuous at any time. The proof is performed by elementary computations.

Norm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups

“…This being true for all x ∈ D(A) we conclude that A = 0. By a result of [25] the same conclusion holds for A = 0 if the range of Q is infinite-dimensional; see also [11] where the special case of a Hilbert space E was considered.…”

“…Here · L (X) denotes the uniform operator norm of the Banach space L (X) of all bounded linear operators on X. For the heat semigroup, which corresponds to the case A = 0, (1.3) was established earlier in [11]. In Section 2 we extend this result to Banach spaces and complement it by showing that (1.3) also holds whenever S(t) = S(s).…”

Time regularity for generalized Mehler semigroups

“…But in general, generalized Mehler semigroups are not analytic in $$C_b(X)$$ nor in $$BUC(X)$$. Even worse, they are not necessarily eventually norm continuous: see [15, 32, 39] for Ornstein–Uhlenbeck semigroups.…”