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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, ∞).

We consider the problem of L p -boundedness of higher order Riesz transforms ∇ m L −1/2 associated to elliptic operators L of order 2m on R D . As an application of the recently solved Kato conjecture, we show ∇ m L −1/2 ∈ L(L p (R D )) for all p ∈ ( 2D 2m+D ∨1 , 2]. This generalizes the result of Auscher and Tchamitchian restricted to the case D ≤ 2m.

Abstract. Let X be a space of homogeneous type and let L be an injective, non-negative, selfadjoint operator on L 2 (X) such that the semigroup generated by −L fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator, acts as a bounded linear operator on the Hardy space H 1 L (X) associated with L whenever F is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hörmander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators satisfying generalized Gaussian estimates in which the required differentiability order is relaxed compared to all known spectral multiplier results.

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Abstract. We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator A in a Banach space E the following assertions are equivalent: (a) A generates a distribution semigroup; (b) the convolution operator δ ⊗ I − δ ⊗ A has a fundamental solution in D (L(E, D)) where D denotes the domain of A supplied with the graph norm and I denotes the inclusion D → E; (c) A generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.

We study linear control systems in infinite-dimensional Banach spaces governed by analytic semigroups. For p ∈ [1, ∞] and α ∈ R we introduce the notion of L p -admissibility of type α for unbounded observation and control operators. Generalising earlier work by Le Merdy [20] and the first named author and Le Merdy [12] we give conditions under which L p -admissibility of type α is characterised by boundedness conditions which are similar to those in the well-known Weiss conjecture. We also study L p -wellposedness of type α for the full system. Here we use recent ideas due to Pruess and Simonett. Our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take Lebesgue and Besov spaces as state space. This extends the considerations in [4] to non-Hilbertian settings and to p = 2.