Hostname: page-component-7dd5485656-bvgqh Total loading time: 0 Render date: 2025-10-30T17:48:09.911Z Has data issue: false hasContentIssue false
  • English
  • Français

A Short Proof of an Interpolation Theorem

Published online by Cambridge University Press:  20 November 2018

Edward Hughes
Edward Hughes*
Affiliation:
Carleton University, Ottawa, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].

Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 127 - 128
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Dunford, N. and Schwartz, J., Linear operators, Interscience, New York, 1963.Google Scholar
2. Heinz, E., Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123 (1951), 415- 438.Google Scholar
3. Heinz, E., On an inequality of linear operators in a Hilbert space, Nat. Acad. Sci. (U.S.A.) 387 (1955), 27-29.Google Scholar
4. Löwner, K., Über monotone Matrix Functioned Math. Z. 38 (1934), 177-216.Google Scholar
5. Turner, R., On monotone operator functions and interpolation, (to appear).Google Scholar
6. Donoghue, W. F., The Interpolation Of Quadratic Norms, Acta Math. 118 (1967) 251-270.Google Scholar
7. Lions, J. L. and Foias, C., Sur certains théorémes d'interpolation, Acta Sci. Math. (Szeged) 22 (1961) 269-282.Google Scholar