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Proceedings of the American Mathematical Society

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

 

 

Smooth, compact operators


Author: Julien Hennefeld
Journal: Proc. Amer. Math. Soc. 77 (1979), 87-90
MSC: Primary 46B20; Secondary 47B05
MathSciNet review: 539636
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Abstract: It is a result of Holub's [Math. Ann. 201 (1973), 157-163], that for T a compact operator on a real Hilbert space, T is smooth $ \Leftrightarrow \left\Vert {T{x_1}} \right\Vert = \left\Vert {T{x_2}} \right\Vert = \left\Vert T \right\Vert$ for some $ \left\Vert {{x_1}} \right\Vert = \left\Vert {{x_2}} \right\Vert = 1$ implies $ {x_1} = \pm \;{x_2}$. We extend this characterization of smooth, compact operators to a large class of Banach spaces, including $ {l_p},{L_p}[0,1]$, and $ d(a,p)$, with $ 1 < p < \infty $. We show that for this same class of Banach spaces, one dimensional, norm one functionals in $ K{(X)^\ast}$ must be extremal. We also present examples of spaces for which Holub's condition does not characterize smooth, compact operators.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0539636-5
Keywords: Compact operator, smooth, reflexive
Article copyright: © Copyright 1979 American Mathematical Society