## Gateaux derivative of $B(H)$ norm

HTML articles powered by AMS MathViewer

- by Dragoljub J. Kečkic̀ PDF
- Proc. Amer. Math. Soc.
**133**(2005), 2061-2067 Request permission

## Abstract:

We prove that for Hilbert space operators $X$ and $Y$, it follows that \[ \lim _{t\to 0^+}\frac {||X+tY||-||X||}t=\frac 1{||X||} \inf _{\varepsilon >0}\sup _{\varphi \in H_\varepsilon ,||\varphi ||=1} \operatorname {Re}\left <Y\varphi ,X\varphi \right >,\] where $H_\varepsilon =E_{X^*X}((||X||-\varepsilon )^2,||X||^2)$. 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

- Theagenis J. Abatzoglou,
*Norm derivatives on spaces of operators*, Math. Ann.**239**(1979), no. 2, 129–135. MR**519008**, DOI 10.1007/BF01420370 - Rajendra Bhatia and Peter emrl,
*Orthogonality of matrices and some distance problems*, Linear Algebra Appl.**287**(1999), no. 1-3, 77–85. Special issue celebrating the 60th birthday of Ludwig Elsner. MR**1662861**, DOI 10.1016/S0024-3795(98)10134-9 - I. C. Gohberg and M. G. Kreĭn,
*Introduction to the theory of linear nonselfadjoint operators*, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR**0246142**, DOI 10.1090/mmono/018 - Robert C. James,
*Orthogonality and linear functionals in normed linear spaces*, Trans. Amer. Math. Soc.**61**(1947), 265–292. MR**21241**, DOI 10.1090/S0002-9947-1947-0021241-4 - Dragoljub J. Kečkić,
*Orthogonality in ${\mathfrak {S}}_1$ and ${\mathfrak {S}}_\infty$ spaces and normal derivations*, J. Operator Theory**51**(2004), no. 1, 89–104. MR**2055806** - P. J. Maher,
*Commutator approximants*, Proc. Amer. Math. Soc.**115**(1992), no. 4, 995–1000. MR**1086335**, DOI 10.1090/S0002-9939-1992-1086335-6 - Salah Mecheri and Messaoud Bounkhel,
*Global minimum and orthogonality in $C_1$-classes*, J. Math. Anal. Appl.**287**(2003), no. 1, 51–60. MR**2010256**, DOI 10.1016/S0022-247X(03)00480-3 - Barry Simon,
*Trace ideals and their applications*, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR**541149**, DOI 10.1007/BFb0064579

## Additional Information

**Dragoljub J. Kečkic̀**- Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16–18, 11000 Beograd, Serbia & Montenegro
- Email: keckic@matf.bg.ac.yu, keckic@EUnet.yu
- 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
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**133**(2005), 2061-2067 - MSC (2000): Primary 46G05, 47L05; Secondary 47A30
- DOI: https://doi.org/10.1090/S0002-9939-05-07746-4
- MathSciNet review: 2137872