Uniform estimates for fractional operators

Abstract

Given a family of closed operators, $\{A(t)\}$, on a Banach space X of class ${\mathcal H}{\mathcal T}$ we consider the question whether the domain of the fractional operators ${\mathcal D}(A(t)^\theta )$, $0<\theta <1$, coincides with the complex interpolation space $[X,{\mathcal D}(A(t))]_\theta$ such that the embedding constants do not depend on the parameter t. Controlling constants in several fundamental theorems on operators with the property of bounded purely imaginary powers, operators admitting an $\mathcal {H}^{\infty }$calculus, and on complex interpolation theory we find conditions such that the above t-independence of embedding constants holds.