Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

Abstract

We prove comparison theorems for the H ∞ -calculus that allow to transfer the property of having a bounded H ∞ -calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ∞ -calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ∞ -calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L pscale. We apply the results to give new proofs on the H ∞ -calculus for elliptic differential operators, including Schrödinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ∞ -calculus in certain Sobolev spaces and that the Stokes operator on bounded domains with ∂ ∈ C 1,1 has a bounded H ∞ -calculus in the Helmholtz scale L p,σ ( ), p ∈ (1, ∞).

$H^\infty $ calculus and dilatations

“…By ω r (A) we denote the infimum over all such ω. It is shown in [11] that every operator A with a bounded H ∞ calculus is almost R-bounded and, furthermore, we have ω r (A) = ω H ∞ (A). Hence we obtain a characterisation for almost R-bounded operators.…”

Maximal $L^p$-Regularity for Stochastic Evolution Equations

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php that A has a good functional calculus. In recent years, a large body of results has been accumulated by many authors which shows that, as a rule of thumb, any "reasonable" elliptic operator of order 2m has such a calculus (see [3,21,22,26,27,28,29,39,41,54,59,71,87] and the references therein); much of the hard analysis goes into proving these ready-to-use results. Moreover, in most of these examples, the trace space D A (1 − …”

Erratum to: Perturbation and interpolation theorems for the H ∞-calculus with applications to differential operators

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In our article [2], some of the general boundedness results in Sects. 7 and 8 for the H ∞ -functional calculus may not be correct as stated since when applying [4, 1.2.4] our proofs use implicitly certain inclusions of interpolation spaces, which were not stated as assumptions and which do not hold in all generality.

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“…As explained in [2] the equality P(Ẋ γ,A ) =Ẏ γ,B is meant in the following sense: the projection P : X → Y , restricted to X ∩Ẋ γ,A = D(A γ ), has a continuous extensionP :Ẋ γ,A →Ẏ γ,B which is surjective. This also implies that P is compatible with the interpolation couples (X,Ẋ γ,A ) and (Y,Ẏ γ,A ).…”

“…Concerning the meaning of (1) and (2) [2] is contained in the following lemma. Then we apply Theorem 1.3.…”