A generalization of a theorem of J. Holub
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- by Yuri Abramovich PDF
- Proc. Amer. Math. Soc. 108 (1990), 937-939 Request permission
Abstract:
We present here a simple proof of the following result: Let $X$ be an arbitrary $C(K)$ or ${L_1}(\mu )$ space and let $T:X \to X$ be an arbitrary linear continuous operator. Then for at least one choice of signs. \[ \left \| {I \pm T} \right \| = 1 + \left \| T \right \|.\] This is a slightly generalized version of a recent result due to J. Holub [4].References
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Y. A. Abramovich, Injective envelopes of normed lattices, Soviet Math. Dokl. 12 (1971), 511-514.
Y. A. Abramovich and K. Schmidt, Daugavet’s equation and orthomorphisms, preprint.
- Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
- James R. Holub, A property of weakly compact operators on $C[0,1]$, Proc. Amer. Math. Soc. 97 (1986), no. 3, 396–398. MR 840617, DOI 10.1090/S0002-9939-1986-0840617-6
- James R. Holub, Daugavet’s equation and operators on $L^1(\mu )$, Proc. Amer. Math. Soc. 100 (1987), no. 2, 295–300. MR 884469, DOI 10.1090/S0002-9939-1987-0884469-8 A. R. Sourour, MR 88j, no. 47037.
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 937-939
- MSC: Primary 47B38; Secondary 47A30
- DOI: https://doi.org/10.1090/S0002-9939-1990-1002149-5
- MathSciNet review: 1002149