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Hahn-Banach operators

Author(s): M. I. Ostrovskii
Journal: Proc. Amer. Math. Soc. 129 (2001), 2923-2930.
MSC (2000): Primary 46B20, 47A20
Posted: February 22, 2001
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Abstract:

We consider real spaces only.

Definition. An operator $T:X\to Y$ between Banach spaces $X$ and $Y$ is called a Hahn-Banach operator if for every isometric embedding of the space $X$into a Banach space $Z$ there exists a norm-preserving extension $\tilde T$ of $T$ to $Z$.

A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces $(X,Y)$ such that there exists a Hahn-Banach operator $T:X\to Y$of rank $k$. The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman.

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Additional Information:

M. I. Ostrovskii
Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
Address at time of publication: Department of Mathematics, The Catholic University of America, Washington, DC 20064
Email: ostrovskii@cua.edu

DOI: 10.1090/S0002-9939-01-06037-3
PII: S 0002-9939(01)06037-3
Keywords: Hahn-Banach theorem, norm-preserving extension, support set
Received by editor(s): February 9, 2000
Posted: February 22, 2001
Communicated by: Jonathan M. Borwein
Copyright of article: Copyright 2001, American Mathematical Society

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