The $H^{\infty}-$ calculus and sums of closed operators

Abstract

Abstract. We develop a very general operator-valued functional calculus for operators with an H ∞ −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H ∞ calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of L p −maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of L 1 −spaces and C(K)−spaces, to obtain some more detailed results in this setting.

Existence of analytic solutions for the classical Stefan problem

“…This is due to the fact that we can a-priori not guarantee that the sum of the power angles of the single operators in L is strictly less than π, which represents the limiting value in the Dore-Venni result. To our operator L we apply a result of Kalton and Weis [36,Theorem 4.4], as demonstrated in the next lemma.…”

Square functions, bounded analytic semigroups, and applications

“…In this subsection, we introduce the concepts of Rboundedness (see [5,8]) and R-sectoriality (see [56], [57], [29]), which now play a fundamental role for the study of sectorial operators and H ∞ functional calculus. Consider a Rademacher sequence (ε k ) k≥1 on a probability space (Σ, P), that is, a sequence of pairwise independent random variables on Σ such that P(ε k = 1) = P(ε k = −1) = 1 2 for any k ≥ 1.…”

“…Theorem 2.7 (Kalton-Weis [29]). Let A be a sectorial operator on X, and assume that X has property (∆).…”

“…We also include a subsection on Rademacher boundedness, a recent notion which now plays a key 192 C. LE MERDY role in the development of the subject (see e.g. [29,30,35,56,57]). Section 3 is devoted to square functions on Hilbert space.…”

Remarks on Functional Calculus for Perturbed First-order Dirac Operators

“…It is known that the former implies the latter in subspaces of L p [14,Theorem 5.3]. But the converse is not known.…”