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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extending into isometries of $ \mathcal{K}(X,Y)$


Author: T. S. S. R. K. Rao
Journal: Proc. Amer. Math. Soc. 134 (2006), 2079-2082
MSC (2000): Primary 47L05, 46B20
Published electronically: January 5, 2006
MathSciNet review: 2215777
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Abstract: In this paper we generalize a result of Hopenwasser and Plastiras (1997) that gives a geometric condition under which into isometries from $ {\mathcal K}(\ell^2)$ to $ {\mathcal L}(\ell^2)$ have a unique extension to an isometry in $ {\mathcal L}({\mathcal L}(\ell^2))$. We show that when $ X$ and $ Y$ are separable reflexive Banach spaces having the metric approximation property with $ X$ strictly convex and $ Y$ smooth and such that $ {\mathcal K}(X,Y)$ is a Hahn-Banach smooth subspace of $ {\mathcal L}(X,Y)$, any nice into isometry $ \Psi_0 :{\mathcal K}(X,Y)\rightarrow {\mathcal L}(X,Y)$ has a unique extension to an isometry in $ {\mathcal L}({\mathcal L}(X,Y))$.


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Additional Information

T. S. S. R. K. Rao
Affiliation: Stat–Math Unit, Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India
Email: tss@isibang.ac.in

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08178-0
PII: S 0002-9939(06)08178-0
Keywords: Isometries, Hahn-Banach smooth spaces
Received by editor(s): November 8, 2004
Received by editor(s) in revised form: February 15, 2005
Published electronically: January 5, 2006
Additional Notes: This work was done under DST-NSF project DST/INT/US(NSF-RPO-0141)/2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society