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 exact K-monotonic, i.e. the condition $f^0 \in \mathcal{A}$ and the inequality $K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0$ , imply g0∈B and ‖g0‖B≤‖f0‖A (K is Peetre’s K-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 that Tf0=g0, 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.
Citation
Yacin Ameur. "The Calderón problem for Hilbert couples." Ark. Mat. 41 (2) 203 - 231, October 2003. https://doi.org/10.1007/BF02390812
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