Fundamental Properties of Fractional Powers of Unbounded Operators in Banach Spaces

We extend the classical theory of operator-valued analytic functions to a wide class oflinear unbounded operators defined in Banach spaces on sets that need not be everywhere dense.We also describe properties of fractional powers of these operators. The class under considerationincludes the Sturm–Liouville differential operator with homogeneous Dirichlet boundaryconditions that acts in a space of continuous functions on a bounded interval.