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Strongly omnipresent operators: general conditions and applications to composition operators

Part of: Spaces and algebras of analytic functions Special classes of linear operators Linear function spaces and their duals Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  09 April 2009

L. Bernal-González ,
M. C. Calderón-Moreno  and
K.-G. Grosse-Erdmann
L. Bernal-González
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Apdo. 1160, Avenida Reina Mercedes, 41080 Sevilla, Spain e-mail: lbernal@us.es, mccm@us.es
K.-G. Grosse-Erdmann
Affiliation:
Fachbereich Mathematik, Fernuniversität Hagen, 58084 Hagen, Germany e-mail: kg.grosse-erdmann@fernuni-hagen.de
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Abstract

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This paper studies the concept of strongly omnipresent operators that was recently introduced by the first two authors. An operator T on the space H(G) of holomorphic functions on a complex domain G is called strongly omnipresent whenever the set of T-monsters is residual in H(G), and a T-monster is a function f such that Tf exhibits an extremely ‘wild’ behaviour near the boundary. We obtain sufficient conditions under which an operator is strongly omnipresent, in particular, we show that every onto linear operator is strongly omnipresent. Using these criteria we completely characterize strongly omnipresent composition and multiplication operators.

Keywords

holomorphic functionT-monsterresidual setstrongly omnipresent operatordense rangelocally dense rangelocally stable operatorcomposition operatorleft-composition operatormultiplication operator

MSC classification

Secondary: 30E10: Approximation in the complex domain 30H05: Bounded analytic functions 46E10: Topological linear spaces of continuous, differentiable or analytic functions 47B38: Operators on function spaces (general)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 335 - 348
Copyright
Copyright © Australian Mathematical Society 2002

References

[1] Ahlfors, L. V., Complex analysis, 3rd edition (McGraw-Hill, London, 1979).Google Scholar
[2] Bernal-González, L., ‘Omnipresent holomorphic operators and maximal cluster sets’, Colloq. Math. 63 (1992), 315–322.CrossRefGoogle Scholar
[3] Bernal-González, L. and Calderón-Moreno, M. C., ‘Holomorphic T-monsters and strongly omnipresent operators’, J. Approx. Theory 104 (2000), 204–219.CrossRefGoogle Scholar
[4] Birkhoff, G. D., ‘Démonstration d'un théorème élémentaire sur les fonctions entièveres’, C. R. Acad. Sci. Paris 189 (1929), 473–475.Google Scholar
[5] Collingwood, E. F. and Lohwater, A. J., The theory of cluster sets (Cambridge University Press, Cambridge, 1966).CrossRefGoogle Scholar
[6] Ehrenpreis, L., ‘Mean periodic functions I’, Amer. J. Math. 77 (1955), 293–328.CrossRefGoogle Scholar
[7] Grosse-Erdmann, K.-G., ‘Holomorphe Monster und universelle Funktionen’, Mitt. Math. Sem. Giessen 176 (1987).Google Scholar
[8] Luh, W., ‘Holomorphic monsters’, J. Approx. Theory 53 (1988), 128–144.CrossRefGoogle Scholar
[9] Luh, W., ‘Multiply universal holomorphic functions’, J. Approx. Theory 89 (1997), 135–155.CrossRefGoogle Scholar
[10] Luh, W. and Martirosian, V. A. and Müller, J., ‘ T-universal functions with lacunary power series’, Acta Sci. Math. (Szeged) 64 (1998), 67–79.Google Scholar
[11] Malgrange, B., ‘Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution’, Ann. Inst. Fourier (Grenoble) 6 (1955/1956), 271–355.CrossRefGoogle Scholar
[12] Noshiro, K., Cluster sets (Springer, Berlin, 1960).CrossRefGoogle Scholar
[13] Rudin, W., Real and Complex Analysis, 2nd edition (Tata McGraw-Hill, Faridabad, 1974).Google Scholar
[14] Schneider, I., ‘Schlichte Funktionen mit universellen Approximationseigenschaften’, Mitt. Math. Sem. Giessen. 230 (1997).Google Scholar