A generalization of a theorem of J. Holub. (English) Zbl 0705.47007
J. Holub proved in Proc. Am. Math. Soc. 97, 396-398 (1986; Zbl 0601.47029) that for an arbitrary continuous linear operator T: \(X\to X\), X being the Banach space C[0,1] of all continuous functions on the unit interval [0,1], we have either \(\| I+T\| =1+\| T\|\), or \(\| I-T\| =1+\| T\|\) (or both). The author generalizes this result for the cases where X is C(K), \(L_ 1(\mu)\), an AL-, or an AM space.
Reviewer: I.Gottlieb
MSC:
| 47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
| 47B38 | Linear operators on function spaces (general) |
| 46B25 | Classical Banach spaces in the general theory |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
Citations:
Zbl 0601.47029References:
| [1] | Y. A. Abramovich, Injective envelopes of normed lattices, Soviet Math. Dokl. 12 (1971), 511-514. · Zbl 0225.46011 |
| [2] | Y. A. Abramovich and K. Schmidt, Daugavet’s equation and orthomorphisms, preprint. · Zbl 0723.46014 |
| [3] | Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. · Zbl 0608.47039 |
| [4] | James R. Holub, A property of weakly compact operators on \?[0,1], Proc. Amer. Math. Soc. 97 (1986), no. 3, 396 – 398. · Zbl 0601.47029 |
| [5] | James R. Holub, Daugavet’s equation and operators on \?\textonesuperior (\?), Proc. Amer. Math. Soc. 100 (1987), no. 2, 295 – 300. · Zbl 0633.47016 |
| [6] | A. R. Sourour, MR 88j, no. 47037. |
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