Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1155602
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $Y$ be a Hausdorff topological space, let $V$ be a system of weights on $Y$, and let $L{V_0}(Y)$ and $L{V_b}(Y)$ 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 $L{V_0}(Y)$ and $L{V_b}(Y)$ 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.

References [Enhancements On Off] (What's this?)

  • M. B. Abrahamse, Multiplication operators, Hilbert space operators (Proc. Conf., Calif. State Univ., Long Beach, Calif., 1977) Lecture Notes in Math., vol. 693, Springer, Berlin, 1978, pp. 17–36. MR 526530
  • Klaus-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 (Proc. Sem., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1978) Lecture Notes in Math., vol. 843, Springer, Berlin, 1981, pp. 93–149. MR 610827
  • S. Machado and J. B. Prolla, The general complex case of the Bernšteĭn-Nachbin approximation problem, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, vii, 193–206 (English, with French summary). MR 500921
  • Leopoldo Nachbin, Silvio Machado, and João B. Prolla, Weighted approximation, vector fibrations and algebras of operators, J. Math. Pures Appl. (9) 50 (1971), 299–323. MR 293397
  • Leopoldo Nachbin, João B. Prolla, and Silvio Machado, Concerning weighted approximation, vector fibrations, and algebras of operators, J. Approximation Theory 6 (1972), 80–89. MR 350425, DOI
  • João Prolla, The approximation property for Nachbin spaces, Approximation theory and functional analysis (Proc. Internat. Sympos. Approximation Theory, Univ. Estadual de Campinas, Campinas, 1977) North-Holland Math. Stud., vol. 35, North-Holland, Amsterdam-New York, 1979, pp. 371–382. MR 553428
  • R. K. Singh and Jasbir Singh Manhas, Multiplication operators on weighted spaces of vector-valued continuous functions, J. Austral. Math. Soc. Ser. A 50 (1991), no. 1, 98–107. MR 1094062
  • R. K. Singh and Jasbir Singh Manhas, Multiplication operators and dynamical systems, J. Austral. Math. Soc. Ser. A 53 (1992), no. 1, 92–102. MR 1164779
  • ---, Operators and dynamical systems on weighted function spaces, preprint.
  • Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B37, 54H20

Retrieve articles in all journals with MSC: 47B37, 54H20

Additional Information

Keywords: Weights, cross-sections, seminorms, vector-valued and operator-valued mappings, multiplication operators, dynamical systems
Article copyright: © Copyright 1993 American Mathematical Society