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October 2003 The Calderón problem for Hilbert couples
Yacin Ameur
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Ark. Mat. 41(2): 203-231 (October 2003). DOI: 10.1007/BF02390812


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 g0B and ‖g0B≤‖f0A (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.


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Yacin Ameur. "The Calderón problem for Hilbert couples." Ark. Mat. 41 (2) 203 - 231, October 2003.


Received: 8 January 2002; Revised: 11 April 2002; Published: October 2003
First available in Project Euclid: 31 January 2017

zbMATH: 1064.46065
MathSciNet: MR2011918
Digital Object Identifier: 10.1007/BF02390812

Rights: 2003 © Institut Mittag-Leffler


Vol.41 • No. 2 • October 2003
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