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 | 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.

**[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-*, J. London Math. Soc (2)

bounded operators**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