On approximation of solutions of operator-differential equations with their entire solutions of exponential type

We consider an equation of the form $y^{\mathrm{\prime }}(t)+Ay(t)=0,t\in [0,\mathrm{\infty })$, 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 $L_2(\mathrm{\Omega })$ \ (the domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is bounded with smooth boundary) and a certain boundary condition.