Exponential type vectors of isometric group
generators

A. Bednarz; O. Lopushansky

By means of exponential type vectors a criterion for operator in a Banach space to generate groups of isometries is established. It is also shown that exponential type vectors of operators with separable spectrum can be completely described by spectral subspaces.

1. Bergh J., Löfström J. Interpolation spaces, Springer-Verlag, 1976.

2. Bieberbach L. Analytische fortsezung, Springer-Verlag, 1956.

3. Gorbachuk M. L., Gorbachuk V. I. On approximation of smooth vectors of closed operator by entire exponential type vectors, Ukrain. Math. J. 47 (1995), № 5, 616–628 (in Ukrainian).

4. Dunford N., Schwartz J. T. Linear operators. Part III, Wiley-Interscience, 1971.

5. Plansherel M., Polya G. Fonctions entiéres et intégrales de Fourier multiples, Comment. Math., Helvetici. I (1937), 224–248; II (1938), 110–163.

6. Levinson N. Gap and density theorems, N.Y., 1940.

7. Lopushansky O. Operator calculus on ultrasmooth vectors, Ukrain. Math. J. 44 (1992), no 4, 502–513.

8. Lopushansky O., Dmytryshyn M. Vectors of exponential type of operators with discrete spectrum Matem. Studii. 9 (1998), no 1, 70–77.

9. Lopushansky O., Dmytryshyn M. Operator calculus on the exponential type vectors of the operator with point spectrum. In book "General Topology in Banach Spaces", Nova Sci. Publ. Huntingon, New York, 2001, 137–145.

10. Lubic J., Macajev V. On operators with separable spectrum Matem. Sbornik, 56 (1962), № 4, 433–468 (in Russian).

11. Radzievsky G. V. On the best approximation and rate of convergence of expansions along operator root vectors Ukrain. Math. J. 49 (1997), № 6, 754–773 (in Russian).

12. Radyno Ja. Exponential type vectors in operator calculus and differential equations Differential Equations, 21 (1985), № 9, 1559–1569 (in Russian).

13. Reed M., Simon B. Methods of modern mathematical physics. V 2. Academic Press, 1975.

	Exponential type vectors of isometric group generators
Author	A. Bednarz, O. Lopushansky Institute of Mathematics, Cracow University of Technology,ul. Warszawska 24, 31–155 Cracow, Poland
Abstract	By means of exponential type vectors a criterion for operator in a Banach space to generate groups of isometries is established. It is also shown that exponential type vectors of operators with separable spectrum can be completely described by spectral subspaces.
Keywords	exponential type vectors, criterion for operator, Banach space, generate groups of isometries, separable spectrum, spectral subspaces
DOI	doi:10.30970/ms.18.1.99-106
Reference	1. Bergh J., Löfström J. Interpolation spaces, Springer-Verlag, 1976. 2. Bieberbach L. Analytische fortsezung, Springer-Verlag, 1956. 3. Gorbachuk M. L., Gorbachuk V. I. On approximation of smooth vectors of closed operator by entire exponential type vectors, Ukrain. Math. J. 47 (1995), № 5, 616–628 (in Ukrainian). 4. Dunford N., Schwartz J. T. Linear operators. Part III, Wiley-Interscience, 1971. 5. Plansherel M., Polya G. Fonctions entiéres et intégrales de Fourier multiples, Comment. Math., Helvetici. I (1937), 224–248; II (1938), 110–163. 6. Levinson N. Gap and density theorems, N.Y., 1940. 7. Lopushansky O. Operator calculus on ultrasmooth vectors, Ukrain. Math. J. 44 (1992), no 4, 502–513. 8. Lopushansky O., Dmytryshyn M. Vectors of exponential type of operators with discrete spectrum Matem. Studii. 9 (1998), no 1, 70–77. 9. Lopushansky O., Dmytryshyn M. Operator calculus on the exponential type vectors of the operator with point spectrum. In book "General Topology in Banach Spaces", Nova Sci. Publ. Huntingon, New York, 2001, 137–145. 10. Lubic J., Macajev V. On operators with separable spectrum Matem. Sbornik, 56 (1962), № 4, 433–468 (in Russian). 11. Radzievsky G. V. On the best approximation and rate of convergence of expansions along operator root vectors Ukrain. Math. J. 49 (1997), № 6, 754–773 (in Russian). 12. Radyno Ja. Exponential type vectors in operator calculus and differential equations Differential Equations, 21 (1985), № 9, 1559–1569 (in Russian). 13. Reed M., Simon B. Methods of modern mathematical physics. V 2. Academic Press, 1975.
Pages	99-106
Volume	18
Issue	1
Year	2002
Journal	Matematychni Studii
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Table of content of issue	html

Exponential type vectors of isometric group generators