  Generalizing a result of Roe \cite{Roe} Strichartz proved in \cite{Str} that
if a doubly-infinite sequence $\{f_k\}$ of functions on $\R^n$ satisfies
$f_{k+1}=\Delta f_k$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and
$x\in \R^n$, then $\Delta f_0(x)= -f_0$. Strichartz also showed that the result
fails for hyperbolic 3-space. This negative result can be indeed extended to
any Riemannian symmetric space of noncompact type. Taking this into account we
shall prove that for all Riemannian symmetric spaces of noncompact type the
theorem actually holds true when uniform boundedness is modified suitably.
