  We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0, \infty)$,
where $A$ is a nonnegative self-adjoint operator in a Hilbert space. We give
direct and inverse theorems on approximation of solutions of this equation with
its entire solutions of exponential type. This establishes a one-to-one
correspondence between the order of convergence to $0$ of the best
approximation of a solution and its smoothness degree. The results are
illustrated with an example, where the operator $A$ is generated by a second
order elliptic differential expression in the space $L_{2}(\Omega)$ \ (the
domain $\Omega \subset \mathbb{R}^{n}$ is bounded with smooth boundary) and a
certain boundary condition.
