Lp and operator norm estimates for the complex time heat operator on homogeneous trees

Let X be a homogeneous tree of degree greater than or equal to three. In this paper we study the complex time heat operator H\u3b6 induced by the natural Laplace operator on X. We prove comparable upper and lower bounds for the Lp norms of its convolution kernel h\u3b6 and derive precise estimates for the Lp-Lr operator norms of H\u3b6 for \u3b6 belonging to the half plane Re \u3b6 65 0. In particular, when \u3b6is purely imaginary, our results yield a description of the mapping properties of the Schr\uf6dinger semigroup on X. \ua91998 American Mathematical Society