Interpolation Spaces

Abstract

AbstractIn the present chapter we examine the connections between functional calculus and interpolation spaces. As an ‘appetiser’, in Section 6.1 we present two central ideas: a model describing the real interpolation spaces (X, D(A)) θ,p using the functional calculus and a theorem of Dore. In Section 6.2 we examine the first of these ideas, proving several representation results for the spaces (X, D(Aα)) θ,p . Then we introduce extrapolation spaces for injective operators (Section 6.3). With the help of these spaces in Section 6.4 we derive two fundamental results (Theorem 6.4.2 and Theorem 6.4.5) that lead to more characterisations of interpolation spaces by functional calculus (Section 6.5.1) and a generalisation of Dore’s theorem (Section 6.5.3). In Section 6.6 we establish all the common properties of fractional domain spaces as intermediate spaces: density, the moment inequality, reiteration. Moreover, we prove the intriguing fact that for an operator A ∈ BIP(X) the fractional domain spaces equal the complex interpolation spaces (Theorem 6.6.9). Finally we characterise growth conditions like supt≥1 ‖tθ B(t + A)−1‖ < ∞ in terms of interpolation spaces (Section 6.7).