On summable sequences in the range of a vector measure
Author:
Cándido Piñeiro
Journal:
Proc. Amer. Math. Soc. 125 (1997), 20732082
MSC (1991):
Primary 46G10; Secondary 47B10
MathSciNet review:
1377003
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let . Among other results, we prove that a Banach space has the property that every sequence lies inside the range of an valued measure if and only if, for all sequences in satisfying that the operator is 1summing, the operator is nuclear, being the conjugate number for . We also prove that, if is an infinitedimensional space for , then can't have the above property for any .
 [AD]
R.
Anantharaman and J.
Diestel, Sequences in the range of a vector measure, Comment.
Math. Prace Mat. 30 (1991), no. 2, 221–235. MR 1122692
(92g:46049)
 [DU]
J.
Diestel and J.
J. Uhl Jr., Vector measures, American Mathematical Society,
Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical
Surveys, No. 15. MR 0453964
(56 #12216)
 [KK]
Igor
Kluvánek and Greg
Knowles, Vector measures and control systems, NorthHolland
Publishing Co., AmsterdamOxford; American Elsevier Publishing Co., Inc.,
New York, 1976. NorthHolland Mathematics Studies, Vol. 20; Notas de
Matemática, No. 58. [Notes on Mathematics, No. 58]. MR 0499068
(58 #17033)
 [LP]
J.
Lindenstrauss and A.
Pełczyński, Absolutely summing operators in
𝐿_{𝑝}spaces and their applications, Studia Math.
29 (1968), 275–326. MR 0231188
(37 #6743)
 [LR]
J. Lindenstrauss and H.P. Rosenthal, The spaces, Israel J. Math. 7 (1969), 325349.
 [P]
Albrecht
Pietsch, Operator ideals, NorthHolland Mathematical Library,
vol. 20, NorthHolland Publishing Co., AmsterdamNew York, 1980.
Translated from German by the author. MR 582655
(81j:47001)
 [Pi1]
Cándido
Piñeiro, Operators on Banach spaces taking
compact sets inside ranges of vector measures, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1031–1040. MR 1110552
(93b:47076), http://dx.doi.org/10.1090/S0002993919921110552X
 [Pi2]
C.
Piñeiro, Banach spaces in which every
𝑝weakly summable sequence lies in the range of a vector
measure, Proc. Amer. Math. Soc.
124 (1996), no. 7,
2013–2020. MR 1307557
(96i:46016), http://dx.doi.org/10.1090/S000299399603242X
 [PR]
C.
Piñeiro and L.
RodríguezPiazza, Banach spaces in which every compact
lies inside the range of a vector measure, Proc. Amer. Math. Soc. 114 (1992), no. 2, 505–517. MR 1086342
(92e:46038), http://dx.doi.org/10.1090/S00029939199210863423
 [Ps1]
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
 [Ps2]
G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS, vol.60, Amer. Math. Soc., Providence, R.I., 1984.
 [T]
Nicole
TomczakJaegermann, BanachMazur distances and finitedimensional
operator ideals, Pitman Monographs and Surveys in Pure and Applied
Mathematics, vol. 38, Longman Scientific & Technical, Harlow;
copublished in the United States with John Wiley & Sons, Inc., New
York, 1989. MR
993774 (90k:46039)
 [AD]
 R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Annales Societatis Mathematicae Polonae, Serie I: Comm. Math. XXX (1991), 221235. MR 92g:46049
 [DU]
 J. Diestel and J.J. Uhl, Jr, Vector measures, Math. Surveys 15, Amer. Math. Soc, Providence, R.I, 1977. MR 56:12216
 [KK]
 I. Kluvanek and G. Knowles, Vector measures and control systems, Math. Stud., vol.20, NorthHolland, 1976. MR 58:17033
 [LP]
 J. Lindenstrauss and A. Pelczynski, Absolutely summing operators in spaces and their applications, Studia Math. 29 (1968), 275326. MR 37:6743
 [LR]
 J. Lindenstrauss and H.P. Rosenthal, The spaces, Israel J. Math. 7 (1969), 325349.
 [P]
 A. Pietsch, Operator Ideals, NorthHolland, 1980. MR 81j:47001
 [Pi1]
 C. Piñeiro, Operators on Banach spaces taking compact sets inside ranges of vector measures, Proc. Amer. Math. Soc. 116 (1992), 10311040. MR 93b:47076
 [Pi2]
 C. Piñeiro, Banach spaces in which every weakly summable sequence lies inside the range of a vector measure, Proc. Amer. Math. Soc., to appear. MR 96i:46016
 [PR]
 C. Piñeiro and L. RodriguezPiazza, Banach spaces in which every compact lies inside the range of a measure, Proc. Amer. Math. Soc. 114 (1992), 505517. MR 92e:46038
 [Ps1]
 G. Pisier, Counterexamples to a conjecture of Grothendieck, Acta Mathematica 151 (1983), 181208. MR 85m:46017
 [Ps2]
 G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS, vol.60, Amer. Math. Soc., Providence, R.I., 1984.
 [T]
 N. TomczakJaegermann, BanachMazur Distances and FiniteDimensional Operator Ideals, vol. 38, Pitman Monographs and Surveys in Pure and Appl. Math., 1989. MR 90k:46039
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
46G10,
47B10
Retrieve articles in all journals
with MSC (1991):
46G10,
47B10
Additional Information
Cándido Piñeiro
Email:
candido@colon.uhu.es
DOI:
http://dx.doi.org/10.1090/S0002993997038173
PII:
S 00029939(97)038173
Received by editor(s):
November 30, 1995
Received by editor(s) in revised form:
January 31, 1996
Communicated by:
Palle E. T. Jorgensen
Article copyright:
© Copyright 1997
American Mathematical Society
