Holomorphic functional calculi of operators, quadratic estimates and interpolation

Abstract

We develop some connections between interpolation theory and the theory of bounded holomorphic functional calculi of operators in Hubert spaces, via quadratic estimates. In particular we show that an operator T of type w has a bounded holomorphic functional calculus if and only if the Hilbert space is the complex interpolation space mid-way between the completion of its domain and of its range. We also characterise the complex interpolation spaces of the domains of all the fractional powers of T, whether or not T has a bounded functional calculus. This treatment extends earlier ones for self-adjoint and maximal accretive operators. This work is motivated by the study of first order elliptic systems which are related to the square root problem for non-degenerate second order operators under boundary conditions on an interval. See our subsequent paper [AMcN].