A new proof of Donoghue's interpolation theorem. Issue 3 (2004)

• Record Type: Journal Article
• Authors: Ameur Ameur, Yacin Yacin
• Other Names: Kaijser Kaijser Sten Sten Academic Editor.
• Is Part Of: Journal of function spaces and applications. Volume 2:Issue 3(2004)
• Page Start: 253
• Page End: 265
• Publication Date: 2004
• Subjects: Interpolation of linear operators, Hilbert space.; Function spaces -- Periodicals; 515.73
• DOI: 10.1155/2004/814683
• Languages: English
• ISSNs: 0972-6802

Abstract

Abstract : We give a new proof and new interpretation of Donoghue's interpolation theorem; for an intermediate Hilbert space H ∗ to be exact interpolation with respect to a regular Hilbert couple H ¯ it is necessary and sufficient that the norm in H ∗ be representable in the form ‖ f ‖ ∗ = ( ∫ [ 0, ∞ ] ( 1 + t − 1 ) K 2 ( t, f ; H ¯ ) 2 d ρ ( t ) ) 1 / 2 with some positive Radon measure ρ on 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.