H ∞-functional calculus and models of Nagy–Foiaş type for sectorial operators

Abstract

We prove that a sectorial operator admits an H ∞ -functional calculus if and only if it has a functional model of Nagy-Foiaş type. Furthermore, we give a concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any sectorial operator by passing to a different norm (the McIntosh square function norm). We also show that this quadratic norm is close to the original one, in the sense that there is only a logarithmic gap between them.

On measuring unboundedness of the H∞-calculus for generators of analytic semigroups

“…Remark 4.4. In [12], Galé, Miana and Yakubovich draw a connection between the H ∞calculus for sectorial operators and the theory of functional models for Hilbert space operators. In addition, they prove a logarithmic gap (as they call it) between the Hilbert space X and XA.…”

“…One can show that the H ∞ (Σ φ )-calculus is bounded if and only if the norm • A is equivalent to the norm of the space X, see [12] and the references therein. In the view of [18,Sec.…”

“…6.4.5], where X (1) and X (−1) are the homogeneous spaces for A. In [12], the logarithmic gap refers to the result that (4.9)…”

“…for all x ∈ Λ1(A) −r X, where Λ1(z) = Log(z) + 2πi (here, Log denotes the principal branch of the logarithm) and where Λ −r 1 (A)X is interpreted as a (dense) subspace of X, see Theorem 2.1 in [12]. We learned from D. Yakubovich that this result can be used to derive estimates of (f eε)(A) of the form in (1.2), which are slightly weaker than our results presented here.…”

“…Square function estimates or quadratic estimates play a crucial role in characterizing bounded H ∞ -calculi for sectorial operators, see [7,12,23,24,27]. On Hilbert spaces this means that for some function g ∈ H ∞ an estimate of the form…”