Holomorphic functional calculi of operators, quadratic estimates and interpolation. (English) Zbl 0903.47011

Auscher, Pascal; McIntosh, Alan; Nahmod, Andrea

Indiana Univ. Math. J. 46, No. 2, 375-403 (1997).

Summary: We develop some connections between interpolation theory and the theory of bounded holomorphic functional calculi of operators in Hilbert spaces, via quadratic estimates. In particular we show that an operator \(T\) of type \(\omega\) has a bounded holomorphic functional calculus if and only if the Hilbert space is the complex interpolation space midway 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 selfadjoint 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 the interval. See our subsequent paper [“The square root problem of Kato in one dimension, and first order elliptic systems”, submitted].

MSC:

47A60

Functional calculus for linear operators

Keywords:

interpolation theory; quadratic estimates; bounded holomorphic functional calculus; fractional powers; selfadjoint and maximal accretive operators; first-order elliptic systems; square root problem for non-degenerate second-order operators