Theorems of Wiman–Valiron type for solutions of parabolic equations

In this paper estimates of Wiman–Valiron type are established for solutions of evolution equations of the form
\begin{equation} u'(t)+A(t)u(t)=0 \end{equation}
in a Hilbert space, where $A(t)$ is a positive definite selfadjoint operator with discrete spectrum. In the case of a constant operator $A$ the results characterize the rate of growth of the function $\|u(t)\|$ as $t\to+0$ in terms of the rate of growth of the Fourier coefficients of the initial data.
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