Abstract
In generalizing a series of known results, the following theorem is proved: If K is a continuous linear operator mapping E0 into F0 and E1 into F1 (where E0, E1 and F0, F1, being ideal spaces, are Banach lattices of functions defined on Ω1 and Ω2 respectively), then for any λ. ε (0, 1) K maps E 1−λ0 E λ1 into [(F ′0 )1−λ(F ′1 )λ]′ and is continuous; for suitably chosen norms in the spacesE 1−λ0 E λ1 and [(F ′0 )1−λ(F ′1 )λ] the norm of K is a logarithmically convex function of λ. Six titles are cited in the bibliography.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
A. P. Calderon, Intermediate spaces and interpolation. Complex method “Matematika” (Collection of translations), 9, No. 3, 56–129 (1966).
P. P. Zabrieko, Nonlinear Integral Operators [in Russian], Proceedings of a seminar on functional analysis, Voronezh, No. 8 (1966).
G. Ya. Lozanovskii, Banach structures of A. P. Calderon, Doklady Akad. Nauk SSSR,172, No. 5, 1018–1020 (1967).
M. A. Krasnosel'skii, P. P. Zabreiko, E. I. Pustyl'nik, and P. E. Sobolevskii, Integral Operators in Spaces of Integrable Functions [in Russian], Moscow (1966).
N. Dunford and J. T. Schwartz, Linear Operators, Wiley (1958).
S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Hyperscales of Banach structures, Doklady Akad. Nauk SSSR,170, No. 2, 265–267 (1966).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 593–598, December, 1967.
The author wishes to thank M. A. Krasnosel'skii, under whose direction he is working.
Rights and permissions
About this article
Cite this article
Zabreiko, P.P. Interpolation theorem for linear operators. Mathematical Notes of the Academy of Sciences of the USSR 2, 853–855 (1967). https://doi.org/10.1007/BF01473465
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01473465