Abstract
This paper describes a class of linear operators A which have the property that ∥I+A∥ = 1 + ∥ A ∥, and which act in the space C of abstract continuous functions. In particular, to such operators, called rammers, are referred completely continuous operators and operators acting in C not only from C, but from some essentially broader space. As an application, it is shown that each linear operator B in C, whose norm is 1 and which is the identity of a subspace of finite deficiency, coincides with the identity on the whole space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
I. K. Daugavet, A property of completely continuous operators in the space C, Uspekhi Matem. Nauk,18, No. 5, 157–161 (1963).
G. Ya. Lozanovskii, Almost integral operators in KB-spaces, Vestn. Leningrad University. Ser. Matem., No. 7, 35–44 (1962).
P. P. Zabreiko, Nonlinear integral operators, Transactions of a seminar in functional analysis, Voronezh, No. 8, 1–148 (1966).
W. A. I. Luxemburg and A. C. Zaanen, Notes on Banach function spaces. I-V, Proc. Nederl. Acad. Sci., Amsterdam, ser. A-66 (1963).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 2, No. 6, 599–604, December, 1967.
The author is grateful to P. P. Zabreiko for doscussion of this note.
Rights and permissions
About this article
Cite this article
Krasnosel'skii, M.A. A class of linear operators in the space of abstract continuous functions. Mathematical Notes of the Academy of Sciences of the USSR 2, 856–858 (1967). https://doi.org/10.1007/BF01473466
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01473466