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Well-bounded operators
on nonreflexive Banach spaces


Authors: Cheng Qingping and Ian Doust
Journal: Proc. Amer. Math. Soc. 124 (1996), 799-808
MSC (1991): Primary 47B40; Secondary 46B10, 46B15, 46B20, 47A60
DOI: https://doi.org/10.1090/S0002-9939-96-03098-5
MathSciNet review: 1301522
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Abstract: Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.


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  • [B] E. Berkson, A Fourier analysis theory of abstract spectral decompositions, Surveys of some recent results in operator theory, Vol. 1 (J.B. Conway and B.B. Morrel, eds), Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 95--132. MR 89i:47026
  • [BBG] H. Benzinger, E. Berkson, and T.A. Gillespie, Spectral families of projections, semigroups, and differential operators, Trans. Amer. Math. Soc. 275 (1983), 431--475. MR 84b:47038
  • [BD] E. Berkson and H.R. Dowson, On uniquely decomposable well-bounded operators, Proc. Amer. Math. Soc. 22 (1971), 339--358. MR 44:5803
  • [BG] E. Berkson and T.A. Gillespie, Absolutely continuous functions of two variables and well-
    bounded operators
    , J. London Math. Soc (2) 30 (1984), 305--321. MR 86c:47044
  • [DdL] I. Doust and R. deLaubenfels, Functional calculus, integral representations, and Banach space geometry, Quaestiones Math. 17 (1994), 161--171. MR 95e:47028
  • [Dow] H.R. Dowson, Spectral theory of linear operators, Academic Press, London, 1978. MR 80c:47022
  • [DQ] I. Doust and Qiu Bozhou, The spectral theorem for well-bounded operators, J. Austral. Math. Soc. Ser. A 54 (1993), 334--351. MR 94i:47048
  • [DS] N. Dunford and J.T. Schwartz, Linear operators, Part I: General Theory, Wiley Interscience, New York, 1958. MR 22:8302
  • [K] I. Kluvánek, Characterization of scalar-type spectral operators, Arch. Math. (Brno) 2 (1966), 153--156. MR 35:2163
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, Springer-Verlag, Berlin, 1977. MR 58:17766
  • [P] G. Pisier, Counterexample to a conjecture of Grothendieck, Acta Math. 151 (1983), 181--208. MR 85m:46017
  • [R1] J.R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. Ser. A 1 (1960), 334--343. MR 23:A3463
  • [R2] ------, On well-bounded operators II, Proc. London Math. Soc. (3) 13 (1963), 613--638. MR 27:5124
  • [Si] I. Singer, Basic sequences and reflexivity of Banach spaces, Studia Math. 21 (1962), 351--369. MR 26:4155
  • [Si2] ------, Bases in Banach spaces II, Springer-Verlag, Berlin, 1981.
  • [Sm] D.R. Smart, Conditionally convergent spectral expansions, J. Austral. Math. Soc. Ser. A 1 (1960), 319--333. MR 23:A3462
  • [Sp] P.G. Spain, On well-bounded operators of type (B), Proc. Edinburgh Math. Soc. (2) 18 (1972), 35--48. MR 47:5648

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Additional Information

Cheng Qingping
Affiliation: Department of Mathematics, Jingzhou Teachers College, Jingzhou, Hubei, People’s Republic of China
Email: i.doust@unsw.edu.au

Ian Doust
Affiliation: School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia
Email: cheng@prodigal.murdoch.edu.au

DOI: https://doi.org/10.1090/S0002-9939-96-03098-5
Keywords: Well-bounded operators, functional calculus, nonreflexive Banach spaces
Received by editor(s): August 29, 1994
Additional Notes: This research was supported by the Australian Research Council.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

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