Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A property of compact operators

Author: Herbert Kamowitz
Journal: Proc. Amer. Math. Soc. 91 (1984), 231-236
MSC: Primary 47B38; Secondary 46E99
MathSciNet review: 740177
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note it is shown that if $ T$ is a compact linear operator on a wide class of Banach spaces of the form $ C(S)$, compact $ S$, or $ {L^1}(S,\Sigma ,\mu )$, then $ \left\Vert {I + T} \right\Vert = 1 + \left\Vert T \right\Vert$. This generalizes similar theorems for the spaces $ C\left[ {0,1} \right]$ and $ {L^1}(0,1)$.

References [Enhancements On Off] (What's this?)

  • [1] V. F. Babenko and S. A. Pičugov, On a property of compact operators in the space of integrable functions, Ukrain. Mat. Zh. 33 (1981), no. 4, 491–492 (Russian). MR 627725
  • [2] I. K. Daugavet, A property of completely continuous operators in the space 𝐶, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 157–158 (Russian). MR 0157225
  • [3] N. Dunford and J. T. Schwartz, Linear operators. Part I, Interscience, New York, 1958.
  • [4] C. A. Hayes and C. Y. Pauc, Derivation and martingales, Springer-Verlag, Berlin and New York, 1970.
  • [5] G. E. Shilov and B. L. Gurevich, Integral, measure and derivative: A unified approach, Revised English edition, translated from the Russian and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0194571

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B38, 46E99

Retrieve articles in all journals with MSC: 47B38, 46E99

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society