Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Gateaux derivative of $B(H)$ norm

Author(s): Dragoljub J. Kecki\`c
Journal: Proc. Amer. Math. Soc. 133 (2005), 2061-2067.
MSC (2000): Primary 46G05, 47L05; Secondary 47A30
Posted: January 25, 2005
MathSciNet review: 2137872
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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 $B(H)$.

References:

1.
T.J. Abatzoglu, Norm derivatives on spaces of operators, Math. Ann. 239(1979) 129-135 MR 0519008 (80d:47067)

2.
R. Bhatia and P. Semrl, Orthogonality of matrices and some distance problems, Linear Algebra Appl., 287(1999), 77-85 MR 1662861 (99k:15042)

3.
I.C. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs Vol. 18 (American Mathematical Society 1969) MR 0246142 (39:7447)

4.
R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61(1947) 265-292 MR 0021241 (9:42c)

5.
D.J. Keckic, Orthogonality in $\mathfrak S_1$ and $\mathfrak S_\infty$ spaces and normal derivations, J. Oper. Theory 51(2004), no. 1, 89-104. MR 2055806

6.
P.J. Maher, Commutator approximants, Proc. Amer. Math. Soc. 115-4(1992) 995-1000 MR 1086335 (92j:47059)

7.
S. Mecheri and M. Bounkhel, Global minimum and Orthogonality in $C_1$-classes, J. Math. Anal. Appl. 287(2003), no. 1, 51-60. MR 2010256 (2004g:47028)

8.
B. Simon, Trace ideals and their applications, London Mathematical Society Lecture Notes Series no. 35 (Cambridge University Press, 1979) MR 0541149 (80k:47048)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46G05, 47L05, 47A30

Retrieve articles in all Journals with MSC (2000): 46G05, 47L05, 47A30

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: 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
Posted: January 25, 2005
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia