Spectral Approach to the Fractional Power of Operator and Its Matrix Approximation

This article is devoted to constructing of fractional powers of operators and their matrix approximations. A key feature of this study is the use of a spectral approach that remains applicable even when the base operator does not generate a semigroup. Our main results include the convergence rate of matrix approximation, derived from resolvent estimates, and a practical algorithm for constructing matrix approximations. The theory is supported by examples.