A theorem of Roe and Strichartz for Riemannian symmetric spaces of noncompact type (Q2878707)

Let \(X\) be a noncompact Riemannian manifold and \(\Delta\) be the Laplace--Beltrami operator on \(X\). Generalizing of a result of \textit{J. Roe} [Math. Proc. Camb. Philos. Soc. 87, 69--73 (1980; Zbl 0463.33002)], \textit{R. S. Strichartz} proved in [Trans. Am. Math. Soc. 338, No. 2, 971--979 (1993; Zbl 0815.35012)] that if a doubly infinite sequence \(\{f_k\}\), \(k\in {\mathbb Z}\), of functions on \(X={\mathbb R}^n\) satisfies \(f_{k+1}=\Delta f_k\) and \(| f_k(x)|\leq M\) for all \(k\in {\mathbb Z}\) and \(x\in{\mathbb R}^n\), then \(\Delta f_0=- f_0\). This result fails for any Riemannian symmetric space of noncompact type. In the paper under review the authors prove that for Riemannian symmetric spaces of noncompact type Strichartz's theorem actually holds true the when the uniform boundedness is suitably modified.