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

 

Minimal numerical-radius extensions of operators


Authors: A. G. Aksoy and B. L. Chalmers
Journal: Proc. Amer. Math. Soc. 135 (2007), 1039-1050
MSC (2000): Primary 41A35; Secondary 41A65
Published electronically: September 18, 2006
MathSciNet review: 2262904
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Abstract: In this paper we characterize minimal numerical-radius extensions of operators from finite-dimensional subspaces and compare them with minimal operator-norm extensions. We note that in the cases $ L^p,\ p = 1, \infty$, and in the case of self-adjoint extensions in $ L^2$, the two extensions and their norms are equal.

We also show that, in the case of $ L^p$, $ 1<p<\infty$, and more generally in the case of the dual space being strictly convex, if the minimal projections with respect to the operator norm and with respect to the numerical radius have equal norms, then the operator norm is $ 1$. An analogous result is also true for an arbitrary extension. Finally, we provide an example of a projection from $ l_3^p$ onto a two-dimensional subspace which is minimal with respect to norm but not with respect to the numerical radius for $ p \ne 1, 2, \infty$, and we determine the minimal numerical-radius projection in this same situation.


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Additional Information

A. G. Aksoy
Affiliation: Department of Mathematics, Claremont McKenna College, Claremont, California 91711
Email: asuman.aksoy@claremontmckenna.edu

B. L. Chalmers
Affiliation: Department of Mathematics, University of California, Riverside, California 92507
Email: blc@math.ucr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08609-6
PII: S 0002-9939(06)08609-6
Keywords: Minimal projections, numerical radius, diagonal extremal pairs
Received by editor(s): October 22, 2005
Published electronically: September 18, 2006
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2006 American Mathematical Society