Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T06:43:02.566Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-smooth extension of Frechet differentiability of the norm with applications to numerical ranges

Published online by Cambridge University Press:  18 May 2009

C. Aparicio ,
F. Ocaña ,
R. Payá  and
A. Rodríguez
C. Aparicio
Affiliation:
Departamento de Teoria de Funciones, Facultad de Ciencias, Universidad de Granada, Spain
F. Ocaña
Affiliation:
Departamento de Teoria de Funciones, Facultad de Ciencias, Universidad de Granada, Spain
R. Payá
Affiliation:
Departamento de Teoria de Funciones, Facultad de Ciencias, Universidad de Granada, Spain
A. Rodríguez
Affiliation:
Departamento de Teoria de Funciones, Facultad de Ciencias, Universidad de Granada, Spain
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The following result in the theory of numerical ranges in Banach algebras is well known (see [3, Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w*-continuous states of . As a consequence of the results in this paper the following

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 2 , July 1986 , pp. 121 - 137
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Berberian, S. K., Lectures in functional analysis and operator theory, (Springer-Verlag, 1974).CrossRefGoogle Scholar
2.Bollobás, B., The numerical range in Banach algebras and complex functions of exponential type, Bull. London Math. Soc. 3 (1971), 27–33.CrossRefGoogle Scholar
3.Bonsall, F. F., and Duncan, J., Numerical ranges of operators on normed spaces and of elements of normed algebras, (London Math. Soc. Lecture Note Series 2, Cambridge, 1971).CrossRefGoogle Scholar
4.Bonsall, F. F., and Duncan, J., Numerical ranges II, (London Math. Soc. Lecture Note Series 10, Cambridge, 1973).CrossRefGoogle Scholar
5.Crabb, M. J., Duncan, J., and McGregor, C. M., Some extremal problems in the theory of numerical ranges, Acta Math. 128 (1972), 123–142.CrossRefGoogle Scholar
6.Diestel, J., Geometry of Banach spaces. Selected topics. Lecture Notes in Mathematics No. 48, (Springer-Verlag, 1975).CrossRefGoogle Scholar
7.Dunford, N., and Schwartz, J. T., Linear operators. Part I, (Interscience Publishers, New York, 1958).Google Scholar
8.Giles, J. R., Gregory, D. A., and Sims, Brailey. Geometrical implications of upper semi-continuity of the duality mapping on a Banach Space, Pacific J. Math. 79 (1978), 99–108.CrossRefGoogle Scholar
9.Martinez, J., Mena, J. F., Payà, R., and Rodríguez, A., An approach to numerical ranges without Banach algebra theory. Illinois J. Math., (to appear).Google Scholar
10.Payà, R., Perez, J., and Rodríguez, A., Noncommutative Jordan C*-algebras. Manuscripta Math. 37 (1982), 87–120.CrossRefGoogle Scholar
11.Rodríguez, A., A Vidav-Palmer theorem for Jordan C*-algebras and related topics. J. London Math. Soc.(2) 22 (1980), 318–332.Google Scholar
12.Rodríguez, A., Non-associative normed algebras spanned by hermitian elements. Proc. London Math. Soc. (3) 47 (1983), 258–274.Google Scholar
13.Shultz, F. W., On normed Jordan algebras which are Banach dual spaces. J. Functional Analysis 31 (1979), 360–376.CrossRefGoogle Scholar