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.