Banach space operators with a boundedH∞ functional calculus

Abstract

In this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when / is holomorphic on a larger sector.We also examine how certain properties of this functional calculus, such as the existence of a bounded H°° functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive L p space, then T has a bounded H°° functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.1991 Mathematics subject classification (Amer. Math. Soc): 47A60.

Quadratic estimates and functional calculi of perturbed Dirac operators

“…Thus {z → (I +inΠ B ) −1 } n is a collection of uniformly bounded functions holomorphic on U. Moreover P 0 B u = lim n→∞ (I +inΠ B(z) ) −1 u for all u ∈ H. (This is proved in a setting similar to ours in [11,Theorem 3.8]; we also prove it as a part of Lemma A.1). The second claim now follows from Lemma 6.3.…”

Regularized functional calculi, semigroups, and cosine functionsfor pseudodifferential operators

“…We will prove (1); it will be clear how the proof would be modified for (2). It follows from Proposition 1.…”

“…The operator i d dx , on L 2 (R), is selfadjoint and thus has a BC(R) functional calculus. However, on L p (R), p = 2, it does not have a BC m (R) functional calculus, for any nonnegative integer m; that is, it is not even generalized scalar (see [2, Lemma 5.3]). …”

Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I

“…Then, it is enough to deal only with / e S(S^) thanks to the considerations from the classical density lemmas, namely, Lemma D from [4] and Lemma 2 from [14]. The Cauchy representations used in the proofs of these density lemmas and in Theorems 4.6 and 4.14 (used in tandem as one rigorous assertion) can be deduced as in these sources because, as noticed above, the bounded // 0O (5^)-functional calculus implies the type co, and we can follow the reasoning from [4,14] The assertion of the theorem will follow from Theorem 5.1 if we check the validity of the conditions of that theorem for T = f(U B ), A, = R(tzT[ B ) for some z e €\S dâ nd sufficiently large /.…”

“…First, let us observe that, similarly to [14], one can substitute Lebesgue spaces L p , p 6 (1, oo) with their closed subspaces Y p . However, in our case, we should require them to form an interpolation scale and support the duality pairing Y* -Y',,<.…”