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The numerical ranges of unbounded operators

Published online by Cambridge University Press:  17 April 2009

J.R. Giles  and
G. Joseph
J.R. Giles
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle, New South Wales.
G. Joseph
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle, New South Wales.
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Abstract

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On a Banach space the numerical range of an unbounded operator has a certain density property in the scalar field. Consequently all hermitian and dissipative operators are bounded. For a smooth or separable or reflexive Banach space the numerical range of an unbounded operator is dense in the scalar field.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 1 , August 1974 , pp. 31 - 36
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Asplund, Edgar, “Fréchet differentiability of convex functions”, Acta Math. 121 (1968), 31–47.CrossRefGoogle Scholar
[2]Bonsall, F.F. and Duncan, J., Numerical ranges of operators on normed spaces and of elements of normed algebras (London Mathematical Society Lecture Note Series, 2. Cambridge University Press, Cambridge, 1971).Google Scholar
[3]Gustafson, Karl, “The Toeplitz-Hausdorff Theorem for linear operators”, Proc. Amer. Math. Soc. 25 (1970), 203–204.Google Scholar
[4]Halmos, Paul R., A Hilbert space problem book (Van Nostrand, Princeton, New Jersey; Toronto, Ontario; London; 1967).Google Scholar
[5]Lindenstrauss, Joram, “On nonseparable reflexive Banach spaces”, Bull. Amer. Math. Soc. 72 (1966), 967–970.Google Scholar
[6]Lumer, G. and Phillips, R.S., “Dissipative operators in a Banach space”, Pacific J. Math. 11 (1961), 679–698.Google Scholar
[7]Mazur, S., “Über konvexe Mengen in linearen normierten Räumen”, Studia Math. 4 (1933), 70–84.CrossRefGoogle Scholar
[8]Phelps, R.R., “Some topological properties of support points of convex sets”, Israel J. Math. 13 (1972), 327–336.CrossRefGoogle Scholar