Fundamental properties of fractional powers of unbounded operators in Banach spaces

In this paper, the classical theory of operator-valued analytic functions is extended to a wide class of linear unbounded operators, defined in Banach spaces on not everywhere dense sets. The properties of fractional powers of the corresponding operators are also established. The class under consideration includes Sturm–Liouville differential operators with homogeneous Dirichlet boundary conditions, acting in spaces of continuous functions on bounded intervals.