Wellbounded operators on nonreflexive Banach spaces
Authors:
Cheng Qingping and Ian Doust
Journal:
Proc. Amer. Math. Soc. 124 (1996), 799808
MSC (1991):
Primary 47B40; Secondary 46B10, 46B15, 46B20, 47A60
MathSciNet review:
1301522
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Abstract: Every wellbounded 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 wellbounded 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 wellbounded operator which is not of type (B). We also prove that on any Banach space, compact wellbounded operators have a simple representation as a combination of disjoint projections.
 [B]
Earl
Berkson, A Fourier analysis theory of abstract spectral
decompositions, Surveys of some recent results in operator theory,
Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech.,
Harlow, 1988, pp. 95–132. MR 958571
(89i:47026)
 [BBG]
Harold
Benzinger, Earl
Berkson, and T.
A. Gillespie, Spectral families of projections,
semigroups, and differential operators, Trans.
Amer. Math. Soc. 275 (1983), no. 2, 431–475. MR 682713
(84b:47038), http://dx.doi.org/10.1090/S00029947198306827134
 [BD]
E.
Berkson and H.
R. Dowson, On uniquely decomposable wellbounded operators,
Proc. London Math. Soc. (3) 22 (1971), 339–358. MR 0288607
(44 #5803)
 [BG]
E.
Berkson and T.
A. Gillespie, Absolutely continuous functions of two variables and
wellbounded operators, J. London Math. Soc. (2) 30
(1984), no. 2, 305–321. MR 771426
(86c:47044), http://dx.doi.org/10.1112/jlms/s230.2.305
 [DdL]
Ian
Doust and Ralph
deLaubenfels, Functional calculus, integral representations, and
Banach space geometry, Quaestiones Math. 17 (1994),
no. 2, 161–171. MR 1281587
(95e:47028)
 [Dow]
H.
R. Dowson, Spectral theory of linear operators, London
Mathematical Society Monographs, vol. 12, Academic Press Inc.
[Harcourt Brace Jovanovich Publishers], London, 1978. MR 511427
(80c:47022)
 [DQ]
Ian
Doust and Bo
Zhou Qiu, The spectral theorem for wellbounded operators, J.
Austral. Math. Soc. Ser. A 54 (1993), no. 3,
334–351. MR 1207718
(94i:47048)
 [DS]
Nelson
Dunford and Jacob
T. Schwartz, Linear Operators. I. General Theory, With the
assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics,
Vol. 7, Interscience Publishers, Inc., New York, 1958. MR 0117523
(22 #8302)
 [K]
Igor
Kluvánek, Characterization of scalartype spectral
operators, Arch. Math. (Brno) 2 (1966),
153–156. MR 0211281
(35 #2163)
 [LT]
Joram
Lindenstrauss and Lior
Tzafriri, Classical Banach spaces. I, SpringerVerlag, Berlin,
1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete,
Vol. 92. MR
0500056 (58 #17766)
 [P]
Gilles
Pisier, Counterexamples to a conjecture of Grothendieck, Acta
Math. 151 (1983), no. 34, 181–208. MR 723009
(85m:46017), http://dx.doi.org/10.1007/BF02393206
 [R1]
J.
R. Ringrose, On wellbounded operators, J. Austral. Math. Soc.
1 (1959/1960), 334–343. MR 0126167
(23 #A3463)
 [R2]
J.
R. Ringrose, On wellbounded operators. II, Proc. London Math.
Soc. (3) 13 (1963), 613–638. MR 0155185
(27 #5124)
 [Si]
I.
Singer, Basic sequences and reflexivity of Banach spaces,
Studia Math. 21 (1961/1962), 351–369. MR 0146635
(26 #4155)
 [Si2]
, Bases in Banach spaces II, SpringerVerlag, Berlin, 1981.
 [Sm]
D.
R. Smart, Conditionally convergent spectral expansions, J.
Austral. Math. Soc. 1 (1959/1960), 319–333. MR 0126166
(23 #A3462)
 [Sp]
P.
G. Spain, On wellbounded operators of type (𝐵), Proc.
Edinburgh Math. Soc. (2) 18 (1972/73), 35–48. MR 0317100
(47 #5648)
 [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. 95132. 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), 431475. MR 84b:47038
 [BD]
 E. Berkson and H.R. Dowson, On uniquely decomposable wellbounded operators, Proc. Amer. Math. Soc. 22 (1971), 339358. 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), 305321. MR 86c:47044
 [DdL]
 I. Doust and R. deLaubenfels, Functional calculus, integral representations, and Banach space geometry, Quaestiones Math. 17 (1994), 161171. 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 wellbounded operators, J. Austral. Math. Soc. Ser. A 54 (1993), 334351. 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 scalartype spectral operators, Arch. Math. (Brno) 2 (1966), 153156. MR 35:2163
 [LT]
 J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, SpringerVerlag, Berlin, 1977. MR 58:17766
 [P]
 G. Pisier, Counterexample to a conjecture of Grothendieck, Acta Math. 151 (1983), 181208. MR 85m:46017
 [R1]
 J.R. Ringrose, On wellbounded operators, J. Austral. Math. Soc. Ser. A 1 (1960), 334343. MR 23:A3463
 [R2]
 , On wellbounded operators II, Proc. London Math. Soc. (3) 13 (1963), 613638. MR 27:5124
 [Si]
 I. Singer, Basic sequences and reflexivity of Banach spaces, Studia Math. 21 (1962), 351369. MR 26:4155
 [Si2]
 , Bases in Banach spaces II, SpringerVerlag, Berlin, 1981.
 [Sm]
 D.R. Smart, Conditionally convergent spectral expansions, J. Austral. Math. Soc. Ser. A 1 (1960), 319333. MR 23:A3462
 [Sp]
 P.G. Spain, On wellbounded operators of type (B), Proc. Edinburgh Math. Soc. (2) 18 (1972), 3548. 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:
http://dx.doi.org/10.1090/S0002993996030985
PII:
S 00029939(96)030985
Keywords:
Wellbounded 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
