A new proof of Donoghue's interpolation theorem

Yacin Ameur

doi:10.1155/2004/814683

Abstract

We give a new proof and new interpretation of Donoghue's interpolation theorem; for an intermediate Hilbert spaceH∗to be exact interpolation with respect to a regular Hilbert coupleH¯it is necessary and sufficient that the norm inH∗be representable in the form‖f‖∗=(∫[0,∞](1+t−1)K2(t,f;H¯)2dρ(t))1/2with some positive Radon measureρon the compactified half-line[0,∞]. The result was re-proved in [1] in the finite-dimensional case. The purpose of this note is to extend the proof given in [1] to cover the infinite-dimensional case. Moreover, the presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ‘Donoghue's Lemma’, which is implicitly used in the proof. Hence we take this opportunity to correct that flaw.

Highlights

With some positive Radon measure ρ on the compactified half-line [0, ∞]
The presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ’Donoghue’s Lemma’, which is implicitly used in the proof
All involutions and inner products will be taken with respect to the norm of H0

Summary

Donoghue’s Theorem 1

To see that h∗ is quasi-concave, we make use of the fact that the space H∗ is exact interpolation with respect to the reversed couple (H1, H0) The latter couple has corresponding operator A−1, and we have the relation f (h∗(A)f, f )0. By Lemma 1.2, our problem reduces to showing that a function h corresponding to an exact interpolation space H∗ is necessarily of the form (1.8). For a function h to belong to P | Λ it is necessary and sufficient that the space n2 (h(λ)) is exact interpolation with respect to Letting ε diminish to 0 gives that n2 (h(λ)) is exact interpolation with respect to n2 (λ)), which finishes the proof of Donoghue’s Theorem 1.

Donoghue’s Theorem 2

Connection to Donoghue’s versions

Some remarks on the Foias–Lions theorem