Multiplication operators and dynamical systems on weighted spaces of cross-sections

Authors:
R. K. Singh and J. S. Manhas

Journal:
Proc. Amer. Math. Soc. **119** (1993), 547-554

MSC:
Primary 47B37; Secondary 54H20

DOI:
https://doi.org/10.1090/S0002-9939-1993-1155602-0

MathSciNet review:
1155602

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Abstract: Let be a Hausdorff topological space, let be a system of weights on , and let and be the weighted locally convex spaces of cross-sections with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper, we characterise the multiplication operators on the spaces and induced by the scalar-valued, the vector-valued, and the operator-valued mappings. A (linear) dynamical system on the weighted spaces of cross-sections is obtained as an application of the theory of the multiplication operators. Many examples are given to illustrate the theory.

**[1]**M. B. Abrahamse,*Multiplication operators*:*Hilbert space operators*, Lecture Notes in Math., vol. 693, Springer-Verlag, New York, 1978, pp. 17-36. MR**526530 (80b:47042)****[2]**K. D. Bierstedt,*The approximation-theoretic localization of Schwartz's approximation property for weighted locally convex function spaces and some examples*, Functional Analysis, Holomorphy and Approximation Theory, Lecture Notes in Math., vol. 843, Springer-Verlag, New York, 1981, pp. 93-149. MR**610827 (82j:46031)****[3]**S. Machado and J. B. Prolla,*The general complex case of the Bernstein-Nachbin approximation problem*, Ann. Inst. Fourier (Grenoble)**28**(1978), 193-206. MR**500921 (81g:46069)****[4]**L. Nachbin, S. Machado, and J. B. Prolla,*Weighted approximation, vector fibrations and algebras of operators*, J. Math. Pures Appl. (9)**50**(1971), 299-323. MR**0293397 (45:2474)****[5]**L. Nachbin, J. B. Prolla, and S. Machado,*Concerning weighted approximation, vector fibrations, and algebras of operators*, J. Approx. Theory**6**(1972), 80-89. MR**0350425 (50:2917)****[6]**J. B. Prolla,*The approximation property for Nachbin spaces, approximation theory and functional analysis*, Proc. Conf. Approximation Theory (Campinas 1977), North-Holland Math. Stud., vol. 35, North-Holland, Amsterdam and New York, 1979, pp. 371-382. MR**553428 (81j:46030)****[7]**R. K. Singh and J. S. Manhas,*Multiplication operators on weighted spaces of vector-valued continuous functions*, J. Austral. Math. Soc. Ser. A**50**(1991), 98-107. MR**1094062 (92f:47031)****[8]**-,*Multiplication operators and dynamical systems*, J. Austral. Math. Soc. Ser. A**53**(1992), 92-102. MR**1164779 (93g:47041)****[9]**-,*Operators and dynamical systems on weighted function spaces*, preprint.**[10]**Walter Rudin,*Functional analysis*, Tata McGraw-Hill, New York, 1973. MR**0365062 (51:1315)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1155602-0

Keywords:
Weights,
cross-sections,
seminorms,
vector-valued and operator-valued mappings,
multiplication operators,
dynamical systems

Article copyright:
© Copyright 1993
American Mathematical Society