A fundamental solution of the parabolic equation on Hilbert space. II. The semigroup property

Abstract

The existence of a family of solution operators { q t : t > 0 } \{ {q_t}:t > 0\} corresponding to a fundamental solution of a second order infinite-dimensional differential equation of the form ∂ u / ∂ t = L u \partial u/\partial t = Lu was previously established by the author. In the present paper, it is established that these operators are nonnegative, and satisfy the condition q s q t = q s + t {q_s}{q_t} = {q_{s + t}} .