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A generalization of a theorem of J. Holub

Author: Yuri Abramovich
Journal: Proc. Amer. Math. Soc. 108 (1990), 937-939
MSC: Primary 47B38; Secondary 47A30
MathSciNet review: 1002149
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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.

$\displaystyle \left\Vert {I \pm T} \right\Vert = 1 + \left\Vert T \right\Vert.$

This is a slightly generalized version of a recent result due to J. Holub [4].

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

  • [1] Y. A. Abramovich, Injective envelopes of normed lattices, Soviet Math. Dokl. 12 (1971), 511-514.
  • [2] Y. A. Abramovich and K. Schmidt, Daugavet's equation and orthomorphisms, preprint.
  • [3] C. D. Aliprantis and O. Burkinshaw, Positive operators, Academic Press, New York, London, 1985. MR 809372 (87h:47086)
  • [4] J. Holub, A property of weakly compact operators on $ C[0,1]$, Proc. Amer. Math. Soc. 97 (1986), 396-398. MR 840617 (87j:47028)
  • [5] -, Daugavet's equation and operators on $ {L_1}(\mu )$, Proc. Amer. Math. Soc. 100 (1987), 295-300. MR 884469 (88j:47037)
  • [6] A. R. Sourour, MR 88j, no. 47037.

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