The Calderón problem for Hilbert couples

Abstract

We prove that intermediate Banach spaces$\mathcal{A}$ and$\mathcal{B}$ with respect to arbitrary Hilbert couples$\bar {H}$ and$\bar {K}$ are exact interpolation if and onlyif they are exactK-monotonic, i.e. the condition$f^0 \in \mathcal{A}$ and the inequality$K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0$, implyg ⁰∈B and ‖g ⁰‖B≤‖f ⁰‖ _A (K is Peetre’sK-functional). It is well known that this property is implied by the following: for each ϱ>1 there exists an operator$T:\bar {H} \to \bar {K}$ such thatTf ⁰=g ⁰, and$K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0$. Verifying the latter property, it suffices to consider the “diagonal case” where$\bar {H} = \bar {K}$ is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ϱ=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.