Order continuity of the transition semigroup in Banach spaces (Q1881532)

Let \(E\) be a separable real Banach space and let \(\text{BUC}(E)\) denote the space of all uniformly continuous and bounded functions on \(E\) with real values. The following problem is considered \[ {\partial u \over {\partial t}} = \sum_{k=1}^\infty \lambda_k{\partial^2u \over {\partial x_k^2} }, \] \[ u(0,\cdot)=f(\cdot)\in \text{BUC}(l^p). \] The operator \(\partial^2\over{\partial x_k^2}\) is the generator of the diffusion semigroup \(S_k(t)\). Let \(\{\lambda_k\}\in l^{p/2}\) be a sequence of positive real numbers and \(S(t)f = \prod_{k\geq 1}S_k(\lambda_k t)f\). It is shown that for each \(t>0\), there exists a finite Boreal measure \(\mu_t\) on \(l^p\) such that \[ (S(t)f)(x) = \int_{l^p}f(x-y)\,d\mu_t(y). \]