A new proof of Donoghue's interpolation theorem
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)K2(t,f;H¯...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2004-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2004/814683 |
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| Summary: | 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)K2(t,f;H¯)2dρ(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. |
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| ISSN: | 0972-6802 |