Reflexivity and order properties of scalar-type spectral operators in locally convex spaces

Dodds, P. G.; de Pagter, B.; Ricker, W.

doi:10.1090/S0002-9947-1986-0814927-7

Reflexivity and order properties of scalar-type spectral operators in locally convex spaces
HTML articles powered by AMS MathViewer

by P. G. Dodds, B. de Pagter and W. Ricker PDF

Trans. Amer. Math. Soc. 293 (1986), 355-380 Request permission

Abstract:

One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space $X$ is reflexive. The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections.

References

Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces, Pure and Applied Mathematics, Vol. 76, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493242
William G. Bade, On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345–360. MR 73954, DOI 10.1090/S0002-9947-1955-0073954-0
E. Berkson and H. R. Dowson, On reflexive scalar-type spectral operators, J. London Math. Soc. (2) 8 (1974), 652–656. MR 350495, DOI 10.1112/jlms/s2-8.4.652
Frits Beukers, Charles B. Huijsmans, and Ben de Pagter, Unital embedding and complexification of $f$-algebras, Math. Z. 183 (1983), no. 1, 131–144. MR 701362, DOI 10.1007/BF01187219
Peter G. Dodds and Ben de Pagter, Orthomorphisms and Boolean algebras of projections, Math. Z. 187 (1984), no. 3, 361–381. MR 757477, DOI 10.1007/BF01161953
Peter G. Dodds and Werner Ricker, Spectral measures and the Bade reflexivity theorem, J. Funct. Anal. 61 (1985), no. 2, 136–163. MR 786620, DOI 10.1016/0022-1236(85)90032-1
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
T. A. Gillespie, Cyclic Banach spaces and reflexive operator algebras, Proc. Roy. Soc. Edinburgh Sect. A 78 (1977/78), no. 3-4, 225–235. MR 473907, DOI 10.1017/S0308210500009963
T. A. Gillespie, Boolean algebras of projections and reflexive algebras of operators, Proc. London Math. Soc. (3) 37 (1978), no. 1, 56–74. MR 482360, DOI 10.1112/plms/s3-37.1.56
C. B. Huijsmans and B. de Pagter, Subalgebras and Riesz subspaces of an $f$-algebra, Proc. London Math. Soc. (3) 48 (1984), no. 1, 161–174. MR 721777, DOI 10.1112/plms/s3-48.1.161
Igor Kluvánek, The extension and closure of vector measure, Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972) Academic Press, New York, 1973, pp. 175–190. MR 0335741
Igor Kluvánek, Conical measures and vector measures, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 1, v, 83–105 (English, with French summary). MR 470173
Igor Kluvánek and Greg Knowles, Vector measures and control systems, North-Holland Mathematics Studies, Vol. 20, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. MR 0499068
D. R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157–165. MR 259064
D. R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294–307. MR 291409
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
W. A. J. Luxemburg, Some aspects of the theory of Riesz spaces, University of Arkansas Lecture Notes in Mathematics, vol. 4, University of Arkansas, Fayetteville, Ark., 1979. MR 568706

Riesz spaces

Spectral theory in locally convex spaces

Robert Pallu de La Barrière, Sur les algèbres d’opérateurs dans les espaces hilbertiens, Bull. Soc. Math. France 82 (1954), 1–52 (French). MR 65822
Christian Rall, Über Boolesche Algebren von Projektionen, Math. Z. 153 (1977), no. 3, 199–217. MR 440395, DOI 10.1007/BF01214474
W. Ricker, On Boolean algebras of projections and scalar-type spectral operators, Proc. Amer. Math. Soc. 87 (1983), no. 1, 73–77. MR 677235, DOI 10.1090/S0002-9939-1983-0677235-6
W. Ricker, A spectral mapping theorem for scalar-type spectral operators in locally convex spaces, Integral Equations Operator Theory 8 (1985), no. 2, 276–293. MR 782620, DOI 10.1007/BF01202816
Helmut H. Schaefer, Spectral measures in locally convex algebras, Acta Math. 107 (1962), 125–173. MR 141992, DOI 10.1007/BF02545784
Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978

Recherches sur les fraction continues

8

9

Cyclic Banach spaces and Banach lattices

14

B. Z. Vulikh, Introduction to the theory of partially ordered spaces, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, 1967. Translated from the Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki. MR 0224522
Bertram Walsh, Structure of spectral measures on locally convex spaces, Trans. Amer. Math. Soc. 120 (1965), 295–326. MR 196503, DOI 10.1090/S0002-9947-1965-0196503-1
A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021