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Complex strict and uniform convexity and hyponormal operators

Published online by Cambridge University Press:  24 October 2008

Kirsti Mattila
Kirsti Mattila
Affiliation:
Department of Mathematics, University of Stockholm, 11385 Stockholm
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Abstract

Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. The (spatial) numerical range of an operator TεB(X) is defined as the set

If V(T) ⊂ ℝ, then T is called hermitian. More about numerical ranges may be found in [8] and [9].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 483 - 493
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Akemann, C. and Russo, B.. Geometry of the unit sphere of a C*-algebra and its dual. Pacific J. Math. 32 (1970), 575–585.CrossRefGoogle Scholar
[2]de Barra, G.. Some algebras of operators with closed convex numerical range. Proc. Roy. Irish Acad. Sect. A 72 (1972), 149–154.Google Scholar
[3]Behrends, E.. Normal operators and multipliers on complex Banach spaces and a symmetry property of L1-predual spaces. Israel J. Math. 47 (1984), 23–28.CrossRefGoogle Scholar
[4]Berberian, S. K.. Extensions of a theorem of Fuglede and Putnam. Proc. Amer. Math. Soc. 10 (1959), 175–182.CrossRefGoogle Scholar
[5]Berkson, E.. Hermitian projections and orthogonality in Banach spaces. Proc. London Math. Soc. (3) 24 (1972), 101–118.CrossRefGoogle Scholar
[6]Berkson, E., Fleming, R. J., Goldstein, J. A. and Jamison, J.. One-parameter groups of isometries on Cp. Rev. Roumaine Math. Pures Appl. 24 (1979), 863–868.Google Scholar
[7]Berkson, E., Fleming, R. J. and Jamison, J.. Groups of isometries on certain ideals of Hilbert space operators. Math. Ann. 220 (1976), 151–156.CrossRefGoogle Scholar
[8]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 University Press, 1971).CrossRefGoogle Scholar
[9]Bonsall, F. F. and Duncan, J.. Numerical Ranges II. London Math. Soc. Lecture Note series 10 (Cambridge University Press, 1973).CrossRefGoogle Scholar
[10]Crabb, M. J. and Spain, P. G.. Commutators and normal operators. Glasgow Math. J. 18 (1977), 197–198.CrossRefGoogle Scholar
[11]Davis, W. J., Garling, D. J. H. and Tomczak-Jaegermann, N.. The complex convexity of quasinormed linear spaces. Preprint.Google Scholar
[12]Dowson, H. R., Gillespie, T. A. and Spain, P. G.. A commutativity theorem for hermitian operators. Math. Ann. 220 (1976), 215–217.CrossRefGoogle Scholar
[13]Garling, D. J. H. and Tomczak-Jaegermann, N.. The cotype and uniform convexity of unitary ideals. Israel J. Math. 45 (1983), 175–197.CrossRefGoogle Scholar
[14]Globevnik, J.. On complex strict and uniform convexity. Proc. Amer. Math. Soc. 47 (1975), 175–178.CrossRefGoogle Scholar
[15]Ho, Y.. Derivations on Hilbert-Schmidt operators. Tamkang J. Math. 6 (1975), 191–195.Google Scholar
[16]Kyle, J.. Numerical ranges of derivations. Proc. Edinburgh Math. Soc. 21 (1978), 33–39.CrossRefGoogle Scholar
[17]Legg, D., Scranton, B. and Ward, J.. Chebyshev subspaces in the space of compact operators. J. Approximation Theory 15 (1975), 326–334.CrossRefGoogle Scholar
[18]Mattila, K.. Normal operators and proper boundary points of the spectra of operators on a Banach space. Ann. Acad. Sci. Fenn. Ser. A I, Math. Dissertationes 19 (1978).Google Scholar
[19]Mattila, K.. Complex strict and uniform convexity and hyponormal operators. Report no. 15 (1983), Dept. Math. Univ. of Stockholm.Google Scholar
[20]McCarthy, C. A.. cp. Israel J. Math. 5 (1967), 249–271.CrossRefGoogle Scholar
[21]Moore, R. L., Rodgers, D. D. and Trent, T. T.. A note on intertwining M-hyponormal operators. Proc. Amer. Math. Soc. 83 (1981), 514–516.Google Scholar
[22]Moore, R. L. and Trent, T. T.. A strict maximum modulus theorem for certain Banach spaces. Monatsh. Math. 92 (1981), 197–201.CrossRefGoogle Scholar
[23]Schatten, R.. Norm, Ideals of Completely Continuous Operators (Springer-Verlag, 1960).CrossRefGoogle Scholar
[24]Shaw, S. Y.. On numerical ranges of generalized derivations and related properties. J. Austral. Math. Soc. 36 (1984), 134–142.CrossRefGoogle Scholar
[25]Takahashi, K.. On the converse of Fuglede-Putnam theorem. Acta Sci. Math. (Szeged) 43 (1981), 123–125.Google Scholar
[26]Thorp, E. and Whitley, R.. The strong maximum modulus theorem for analytic functions into a Banach space. Proc. Amer. Math. Soc. 18 (1967), 640–646.CrossRefGoogle Scholar
[27]Williams, J. P.. Finite operators. Proc. Amer. Math. Soc. 26 (1970), 129–136.CrossRefGoogle Scholar