Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Spectral measures, boundedly $ \sigma$-complete Boolean algebras and applications to operator theory


Author: Werner J. Ricker
Journal: Trans. Amer. Math. Soc. 304 (1987), 819-838
MSC: Primary 47D30; Secondary 47B40
DOI: https://doi.org/10.1090/S0002-9947-1987-0911097-2
MathSciNet review: 911097
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A systematic study is made of spectral measures in locally convex spaces which are countably additive for the topology of uniform convergence on bounded sets, briefly, the bounded convergence topology. Even though this topology is not compatible for the duality with respect to the pointwise convergence topology it turns out, somewhat surprisingly, that the corresponding $ {L^1}$-spaces for the spectral measure are isomorphic as vector spaces. This fact, together with I. Kluvanek's notion of closed vector measure (suitably developed in our particular setting) makes it possible to extend to the setting of locally convex spaces a classical result of W. Bade. Namely, it is shown that if $ B$ is a Boolean algebra which is complete (with respect to the bounded convergence topology) in Bade's sense, then the closed operator algebras generated by $ B$ with respect to the bounded convergence topology and the pointwise convergence topology coincide.


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

  • [1] W. G. Bade, On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345-359. MR 0073954 (17:513d)
  • [2] P. G. Dodds and W. Ricker, Spectral measures and the Bade reflexivity theorem, J. Funct. Anal. 61 (1985), 136-163. MR 786620 (86i:47042)
  • [3] P. G. Dodds, B. De Pagter and W. Ricker, Reflexivity and order properties of scalar-type spectral operators in locally convex spaces, Trans. Amer. Math. Soc. 293 (1986), 355-380. MR 814927 (87d:47046)
  • [4] N. Dunford and J. T. Schwartz, Linear operators III: Spectral operators, Interscience Publishers, New York, 1971.
  • [5] I. Kluvanek and G. Knowles, Vector measures and control systems, North-Holland, Amsterdam, 1976. MR 0499068 (58:17033)
  • [6] I. Kluvanek, The range of a vector-valued measure, Math. Systems Theory 7 (1973), 44-54. MR 0322131 (48:495)
  • [7] -, Conical measures and vector measures, Ann. Inst. Fourier (Grenoble) 27(1) (1977), 83-105. MR 0470173 (57:9936)
  • [8] G. Köthe, Topological vector spaces I, Grundlehren Math. Wiss., No. 159, Springer-Verlag, Berlin, Heidelberg, 1983 (2nd printing).
  • [9] -, Topological vector spaces II, Grundlehren Math. Wiss., No. 237, Springer-Verlag, New York, 1979. MR 551623 (81g:46001)
  • [10] S. Okada and W. Ricker, Uniform operator $ \sigma $-additivity of indefinite integrals induced by scalar-type spectral operators, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 141-146. MR 824214 (87f:47048)
  • [11] W. Ricker, On Boolean algebras of projections and scalar-type spectral operators, Proc. Amer. Math. Soc. 87 (1983), 73-77. MR 677235 (84b:47040)
  • [12] -, Closed spectral measures in Fréchet spaces, Internat. J. Math. Math. Sci. 7 (1984), 15-21. MR 743820 (85m:46040)
  • [13] -, Criteria for closedness of vector measures, Proc. Amer. Math. Soc. 91 (1984), 75-80. MR 735568 (85c:28005)
  • [14] -, A spectral mapping theorem for scalar-type spectral operators in locally convex spaces, Integral Equations Operator Theory 8 (1985), 276-288. MR 782620 (86j:47054)
  • [15] A. Shuchat, Vector measures and scalar operators in locally convex spaces, Michigan Math. J. 24 (1977), 303-310. MR 0493497 (58:12497)
  • [16] F. Treves, Topological vector spaces, distributions and kernels, Academic Press, New York, 1967. MR 0225131 (37:726)
  • [17] B. Walsh, Structure of spectral measures on locally convex spaces, Trans. Amer. Math. Soc. 120 (1965), 295-326. MR 0196503 (33:4690)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D30, 47B40

Retrieve articles in all journals with MSC: 47D30, 47B40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0911097-2
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society