|
Well-bounded operators on nonreflexive Banach spaces
Author(s):
Cheng
Qingping;
Ian
Doust
Journal:
Proc. Amer. Math. Soc.
124
(1996),
799-808.
MSC (1991):
Primary 47B40;
Secondary 46B10, 46B15, 46B20, 47A60
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
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.
References:
- [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
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(1991):
47B40,
46B10, 46B15, 46B20, 47A60
Retrieve articles in all Journals with MSC
(1991):
47B40,
46B10, 46B15, 46B20, 47A60
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:
10.1090/S0002-9939-96-03098-5
PII:
S 0002-9939(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
Copyright of article:
Copyright
1996,
American Mathematical Society
Forward Citation(s): Information for authors on submitting citations The following works have cited this article Ian Doust; T. A. Gillespie, An example in the theory of $AC$-operators, Proc. Amer. Math. Soc. 129 (2001), 1453-1457. (English)
|
Comments: webmaster@ams.org