  In this paper, we define an operator function as a series of operators
corresponding to the Taylor series representing the function of the complex
variable. In previous papers, we considered the case when a function has a
decomposition in the Laurent series with the infinite principal part and finite
regular part. Our central challenge is to improve this result having considered
as a regular part an entire function satisfying the special condition of the
growth regularity. As an application we consider an opportunity to broaden the
conditions imposed upon the second term not containing the time variable of the
evolution equation in the abstract Hilbert space.
