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

 

Gateaux derivative of $B(H)$ norm


Author: Dragoljub J. Kecki\`c
Journal: Proc. Amer. Math. Soc. 133 (2005), 2061-2067
MSC (2000): Primary 46G05, 47L05; Secondary 47A30
Published electronically: January 25, 2005
MathSciNet review: 2137872
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Abstract: We prove that for Hilbert space operators $X$ and $Y$, it follows that

\begin{displaymath}\lim_{t\to0^+}\frac{\vert\vert X+tY\vert\vert-\vert\vert X\ve... ...i\vert\vert=1} \operatorname{Re}\left<Y\varphi,X\varphi\right>,\end{displaymath}

where $H_\varepsilon=E_{X^*X}((\vert\vert X\vert\vert-\varepsilon)^2,\vert\vert X\vert\vert^2)$. Using the concept of $\varphi$-Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in $B(H)$, and to give an easy proof of the characterization of smooth points in $B(H)$.


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Additional Information

Dragoljub J. Kecki\`c
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16–18, 11000 Beograd, Serbia & Montenegro
Email: keckic@matf.bg.ac.yu, keckic@EUnet.yu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07746-4
PII: S 0002-9939(05)07746-4
Keywords: Gateaux derivative, orthogonality, smoothness
Received by editor(s): February 3, 2004
Received by editor(s) in revised form: March 7, 2004
Published electronically: January 25, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.