Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T05:20:28.921Z Has data issue: false hasContentIssue false
  • English
  • Français

U-Operators

Part of: Integral, integro-differential, and pseudodifferential operators Miscellaneous topics of analysis in the complex domain Special classes of linear operators General theory of linear operators Operator theory: Ordinary differential operators

Published online by Cambridge University Press:  09 April 2009

L. Bernal-González  and
J. A. Prado-Tendero
L. Bernal-González
Affiliation:
Departmento de Análisis Matemático, Facultad de Mathemáticas, Apdo. 1160, Avenida Reina Mercedes, 41080 Sevilla, Spain e-mail: [email protected], [email protected]
J. A. Prado-Tendero
Affiliation:
Departmento de Análisis Matemático, Facultad de Mathemáticas, Apdo. 1160, Avenida Reina Mercedes, 41080 Sevilla, Spain e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

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.

Inspired by a statement of W. Luh asserting the existence of entire functions having together with all their derivatives and antiderivatives some kind of additive universality or multiplicative universality on certain compact subsets of the complex plane or of, respectively, the punctured complex plane, we introduce in this paper the new concept of U-operators, which are defined on the space of entire functions. Concrete examples, including differential and antidifferential operators, composition, multiplication and shift operators, are studied. A result due to Luh, Martirosian and Müller about the existence of universal entire functions with gap power series is also strengthened.

Keywords

universal functiongap seriescomposition operatordifferential operatorintegral operatorTaylor shiftU-operator

MSC classification

Secondary: 30E10: Approximation in the complex domain 47A16: Cyclic vectors, hypercyclic and chaotic operators 47B33: Composition operators 47B38: Operators on function spaces (general) 47E05: Ordinary differential operators (should also be assigned at least one other classification number in section 47) 47G10: Integral operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 1 , February 2005 , pp. 59 - 90
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bernal-González, L., ‘Omnipresent holomorphic operators and maximal cluster sets’, Colloq. Math. 63 (1992), 315–322.CrossRefGoogle Scholar
[2]Bernal-González, L., ‘Plane sets having dense holomorphic images’, Rev. Roumaine Math. Pures Appl. 40 (1995), 567–569.Google Scholar
[3]Bernal-González, L., ‘Universal functions for Taylor shifts’, Complex Variables 31 (1996), 121–129.Google Scholar
[4]Bernal-González, L., ‘Hypercyclic sequences of differential and antidifferential operators’, J. Approx. Theory 96 (1999), 323–337.CrossRefGoogle Scholar
[5]Bernal-González, L., ‘Densely hereditarily hypercyclic sequences and large hypercyclic manifolds’, Proc. Amer. Math. Soc. 127 (1999), 3279–3285.CrossRefGoogle Scholar
[6]Bernal-González, L. and Calderón-Moreno, M. C., ‘Holomorphic T-monsters and strongly omnipresent operators’, J. Approx. Theory 104 (2000), 204–219.CrossRefGoogle Scholar
[7]Bernal-González, L. and Calderón-Moreno, M. C., ‘Operators with dense images everywhere’, J. Math. Anal. Appl. 263 (2001), 95–109.CrossRefGoogle Scholar
[8]Bernal-González, L. and Calderón-Moreno, M. C., ‘Dense linear manifolds of monsters’, J. Approx. Theory 119 (2002), 156–180.CrossRefGoogle Scholar
[9]Bernal-González, L., Calderón-Moreno, M. C. and Grosse-Erdmann, K. G., ‘Strongly omnipresent operators: general conditions and applications to composition operators’, J. Austral. Math. Soc. 72 (2002), 335–348.CrossRefGoogle Scholar
[10]Bernal-González, L., Calderón-Moreno, M. C. and Grosse-Erdmann, K. G., ‘Strongly omnipresent integral operators’, Integral Equations Operator Theory 44 (2002), 397–409.CrossRefGoogle Scholar
[11]Bernal-González, L. and Montes-Rodríguez, A., ‘Universal functions for composition operators’, Complex Variables 27 (1995), 47–56.Google Scholar
[12]Birkhoff, G. D., ‘Démonstration d'un théorème élémentaire sur les fonctions entières’, C. R. Acad. Sci. Paris 189 (1929), 473–475.Google Scholar
[13]Boas, R. P., Entire functions (Academic Press, New York, 1954).Google Scholar
[14]Calderón-Moreno, M. C., ‘Holomorphic differential operators and plane sets with dense images’, Complex Variables 47 (2002), 167–176.Google Scholar
[15]Calderón-Moreno, M. C. and Müller, J., ‘Universal holomorphic and harmonic functions with additional properties’, Acta Math. Hungar. 105 (2004), 1–15.CrossRefGoogle Scholar
[16]Ehrenpreis, L., ‘Mean periodic functions I’, Amer. J. Math. 77 (1955), 293–328.CrossRefGoogle Scholar
[17]Godefroy, G. and Shapiro, J. H., ‘Operators with dense, invariant, cyclic vectors manifolds’, J. Funct. Anal. 98 (1991), 229–269.CrossRefGoogle Scholar
[18]Grosse-Erdmann, K. G., ‘Universal families and hypercyclic vectors’, Bull. Amer. Math. Soc.(N.S.) 36 (1999), 345–381.CrossRefGoogle Scholar
[19]Grosse-Erdmann, K. G., ‘Hypercyclic and chaotic weighted shifts’, Studio Math. 139 (2000), 47–68.CrossRefGoogle Scholar
[20]Heins, M., ‘On the number of 1–1 directly conformed maps which a multiply-connected plane region of finite connectivity p (>2) admits onto itself’, Bull. Amer. Math. Soc. 52 (1946), 454–457.CrossRefGoogle Scholar
[21]Hille, E., Analytic function theory, II (Chelsea, New York, 1987).Google Scholar
[22]Luh, W., ‘Holomorphic monsters’, J. Approx. Theory 53 (1988), 128–144.CrossRefGoogle Scholar
[23]Luh, W., ‘Universal functions and conformal mappings’, Serdica 19 (1993), 161–166.Google Scholar
[24]Luh, W., ‘Entire functions with various universal properties’, Complex Variables 31 (1996), 87–96.Google Scholar
[25]Luh, W., ‘Multiply universal holomorphic functions’, J. Approx. Theory 89 (1997), 135–155.CrossRefGoogle Scholar
[26]Luh, W., Martirosian, V. A. and Müller, J., ‘T-universal functions with lacunary power series’, Acta Sci. Math.(Szeged) 64 (1998), 67–79.Google Scholar
[27]Luh, W., Martirosian, V. A. and Müller, J., ‘Universal entire functions with gap power series’, Indag. Math., N.S. 9 (1998), 529–536.CrossRefGoogle Scholar
[28]Luh, W., Martirosian, V. A. and Müller, J., ‘Restricted T-universal functions’, J. Approx. Theory 114 (2002), 201–213.CrossRefGoogle Scholar
[29]Maclane, G. R., ‘Sequences of derivatives and normal families’, J. Analyse Math. 2 (1952/1953), 72–87.CrossRefGoogle Scholar
[30]Malgrange, B., ‘Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution’, Ann. Institut Fourier(Grenoble) 6 (1955/1956), 271–355.CrossRefGoogle Scholar
[31]Mathew, V., ‘A note on hypercyclic operators on the space of entire sequences’, Indian J. Pure Appl. Math. 25 (1994), 1181–1184.Google Scholar
[32]Pólya, G., ‘Untersuchungen über Lücken und Singularitäten von Potenzreihen(1. Mitteilung)’, Math. Z. 29 (1929), 549–640.CrossRefGoogle Scholar
[33]Rudin, W., Real and complex analysis, 3rd edition (McGraw-Hill, New York, 1987).Google Scholar
[34]Schneider, I., ‘Schlichte Funktionen mit universellen Approximationseigenschaften’, Mitt. Math. Sem. Giessen 230 (1997).Google Scholar
[35]Seidel, W. and Walsh, J. L., ‘On approximation by Euclidean and non-Euclidean translation of analytic functions’, Bull. Amer. Math. Soc. 42 (1941), 916–920.CrossRefGoogle Scholar
[36]Tenthoff, R., Universelle holomorphe Funktionen mit vorgegeben Approximationswegen (Shaker, Aachen, 2000).Google Scholar
[37]Zappa, P., ‘On universal holomorphic functions’, Boll. Un. Mat. Ital. A (7) 2 (1988), 345–352.Google Scholar