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

### Abstract

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.

**Key words:** Hilbert and Banach spaces, differential-operator equation, weak solution, $C_{0}$-semigroup of linear operators, entire vector-valued function, entire vector-valued function of exponential type, the best approximation, direct and inverse theorems of the approximation theory.