On analytic solutions of the heat equation with an operator coefficient

Let $A$ be a bounded linear operator on a Banach space and $g$ a vector-valued function analytic on a neighborhood of the origin of $\mathbb R$. We obtain conditions for the existence of analytic solutions for the Cauchy problem
$$ \begin{cases} \dfrac{\partial u}{\partial t}=A^2\dfrac{\partial^2u}{\partial x^2},\\u(0,x)=g(x). \end{cases} $$
Moreover, we consider a representation of the solution of this problem as a Poisson integral and investigate the Cauchy problem for the corresponding nonhomogeneous equation. Bibl. – 22 titles.