$H^∞$ functional calculus in real interpolation spaces

Giovanni Dore

ArticleOriginal scientific text

Title

$H^{\infty}$ functional calculus in real interpolation spaces

Authors ¹

Affiliations

Dipartimento di Matematica, Università di Bologna, Piazza di Porta, S. Donato 5, 40127 Bologna, Italy

Abstract

Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and

∥ λ {(λ I - A)}^{- 1} ∥

is bounded outside every larger sector) and has a bounded inverse, then A has a bounded

H^{\infty}

functional calculus in the real interpolation spaces between X and the domain of the operator itself.

K. N. Boyadzhiev and R. J. deLaubenfels, $H^{\infty}$ functional calculus for perturbations of generators of holomorphic semigroups, Houston J. Math. 17 (1991), 131-147.
K. N. Boyadzhiev and R. J. deLaubenfels, Semigroups and resolvents of bounded variation, imaginary powers and $H^{\infty}$ functional calculus, Semigroup Forum 45 (1992), 372-384.
M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded $H^{\infty}$ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51-89.
G. Dore and A. Venni, Some results about complex powers of closed operators, J. Math. Anal. Appl. 149 (1990), 124-136.
R. deLaubenfels, A holomorphic functional calculus for unbounded operators, Houston J. Math. 13 (1987), 545-548.
R. deLaubenfels, Unbounded holomorphic functional calculus and abstract Cauchy problems for operators with polynomially bounded resolvents, J. Funct. Anal. 114 (1993), 348-394.
A. McIntosh, Operators which have an $H_{\infty}$ functional calculus, in: Miniconference on Operator Theory and Partial Differential Equations (Macquarie University, Ryde, N.S.W., September 8-10, 1986), B. Jefferies, A. McIntosh and W. Ricker (eds.), Proc. Centre Math. Anal. Austral. Nat. Univ. 14, Austral. Nat. Univ., Canberra, 1986, 210-231.
A. McIntosh and A. Yagi, Operators of type ω without a bounded $H_{\infty}$ functional calculus, in: Miniconference on Operators in Analysis (Macquarie University, Sydney, N.S.W., September 25-27, 1989), I. Doust, B. Jefferies, C. Li and A. McIntosh (eds.), Proc. Centre Math. Anal. Austral. Nat. Univ. 24, Austral. Nat. Univ., Canberra, 1989, 159-172.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

Title

H∞ functional calculus in real interpolation spaces

Affiliations

Abstract

Bibliography

$H^{\infty}$ functional calculus in real interpolation spaces