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Imbedding elements whose numerical range has a vertex at zero in holomorphic semigroups

Published online by Cambridge University Press:  20 January 2009

J. Martinez-Moreno  and
A. Rodriguez-Palacios
J. Martinez-Moreno
Affiliation:
Universidad de GranadaFacultad de CienciasDepto. de Teoria de FuncionesGranada (Spain)
A. Rodriguez-Palacios
Affiliation:
Universidad de GranadaFacultad de CienciasDepto. de Teoria de FuncionesGranada (Spain)
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If a is an element of a complex unital Banach algebra whose numerical range is confined to a closed angular region with vertex at zero and angle strictly less than π, we imbed a in a holomorphic semigroup with parameter in the open right half plane.

There has been recently a great development in the theory of semigroups in Banach algebras (see [6]), with attention focused on the relation between the structure of a given Banach algebra and the existence of continuous or holomorphic non-trivial semigroups with certain properties with range in this algebra. The interest of this paper arises from the fact that we relate in it, we think for the first time, this new point of view in the theory of Banach algebras with the already classic one of numerical ranges [2,3]. The proofs of our results use, in addition to some basic ideas from numerical ranges in Banach algebras, the concept of extremal algebra Ea(K) of a compact convex set K in ℂ due to Bollobas [1] and concretely the realization of Ea(K) achieved by Crabb, Duncan and McGregor [4].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 1 , February 1985 , pp. 91 - 95
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

1.Bollobas, B., The numerical range in Banach algebras and complex functions of exponential type, Bull. London Math. Soc. 3 (1971), 27–33.CrossRefGoogle Scholar
2.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
3.Bonsall, F. F. and Duncan, J., Numerical ranges II (London Math. Soc. Lecture Note Series 10, Cambridge, 1973).CrossRefGoogle Scholar
4.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
5.Hille, S. and Phillips, R. S., Functional analysis and semigroups (Revised Ed., American Math. Soc., 1974).Google Scholar
6.Sinclair, A. M., Continuous Semigroups in Banach Algebras (London Math. Soc. Lecture note Series 63, Cambridge, 1982).CrossRefGoogle Scholar