A discrete framework for the interpolation of Banach spaces

Abstract

We develop a discrete framework for the interpolation of Banach spaces, which contains e.g. the well-known real and complex interpolation methods, but also more exotic methods like the ±-method, the Radamacher interpolation method and the ℓ p -interpolation method, as concrete examples. Our method is based on a sequential structure imposed on a Banach space, which allows us to deduce properties of interpolation methods from properties of sequential structures.Our framework has both a formulation modelled after the real and the complex interpolation methods. This allows us to extend results previously known for either the real or the complex interpolation method to all interpolation methods that fit into our framework. As applications of this observation we prove abstract Stein interpolation and the interpolation of intersections for all methods that fit into our framework.

Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications

“…To verify that [52,Corollary 7.8] is applicable to S −α0 one needs Lemma A.2 and that the spaces Ḋ(T α ) have non-trivial type, which is equivalent to B-convexity, see [44,Proposition 7.6.8], this follows since Ḋ(T θ ) are isomorphic to X, compare [52, Section 2]. Therefore S −α0 has a bounded H ∞ -calculus for all θ ∈ (0, 1) on Ḋ(S 0 −α0 , Ḋ(S 1 −α0 ) θ = Ḋ(T −α ), Ḋ(T 1−α ) θ = [ Ḋ(T −α ), Ḋ(T 1−α )] θ = Ḋ(T θ−α ), here •, • θ denotes the Rademacher interpolation functor introduced in [52, Section 7] (see also [51,76]), and one has applied [52,Theorem 7.4] to T which guarantees that here the complex and the Rademacher interpolation scale agree for T . So, S −α0 | Ḋ(T −α 0 +θ ) has a bounded H ∞ -calculus for all θ ∈ (0, 1) on Ḋ(T −α0+θ ).…”