Let X = G/K be a symmetric space of noncompact type, L be the Laplace-Beltrami operator on X, and b the bottom of its spectrum. In this paper we study the Lp-Lq mapping properties of several families of operators naturally associated with L: θ-heat semigroup, H_{t,θ) = exp(tL−θb), complex powers of resolvent operator, H^α_θ=(L−θb)^{−α/2}, and S^α_θ = (L−θb)^{−α/2} exp(i(L−θb)), where 0≤θ≤1, Reα≥ 0, closely related to the Cauchy problem for the Schrodinger operator on X. The techniques mix harmonic analysis on semisimple Lie groups (Plancherel measure, c-function) and functional analysis (interpolation, semigroup theory). One of the contribution in this paper is to give precise estimates for the Lp-Lq operator norm of H_{t,θ) for large time t on all noncompact symmetric spaces. These estimates show that the interpolation and extrapolation methods of ultracontractivitywhich work well for small t or on groups of polynomial growth are not applicable here; in particular, log(|||H_{t,θ)|||_{p,q}) does not depend linearly on 1/p and 1/q.