On the norm continuity of transition semigroups in Hilbert spaces
Let \(H\) be a separable Hilbert space, \((e_n)_{n\in\mathbb{N}}\) a complete orthonormal system, \(\text{UBC}(H)=\{\varphi: H\to\mathbb{R}\) uniformly continuous and bounded functions\}, \(\lambda_n\in ]0,\infty[\), \(\sum_{n\in\mathbb{N}} \lambda_n<+\infty\), \((P_t)_{t>0}\) the transition semigroup solution of \(\partial_1u(t,x)={1\over 2} \sum_{n\in\mathbb{N}} \lambda_n\partial_{n+1}\partial_{n+1}u(t,x)\), \(t\geq 0\), \(x\in H\), \(u(0,x)= \varphi(x)\), \(x\in H\), where \(\varphi\in\text{UBC}(H)\) and \(\partial_{n+1}u(t,.)= \lim_{h\to 0} (u(t,.+he_n)-u(t,.))/h\). The authors prove that \(t\in]0,+\infty[\mapsto P_t\in L(\text{UBC}(H))\) is not continuous for the operator norm on UBC\((H)\).
scientific article; zbMATH DE number 1145381