|
Matrix summability and uniform convergence of series
Authors:
Antonio Aizpuru, Francisco J. García-Pacheco and Consuelo Pérez-Eslava
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3571-3579
MSC (2000):
Primary 46B15, 46B25, 46B20
Posted:
June 21, 2007
MathSciNet review:
2336572
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Some classical results about uniform convergence of unconditionally convergent series are generalized to weakly unconditionally Cauchy series by means of the matrix summability method for regular matrices.
- 1.
A.
Aizpuru and J.
Pérez-Fernández, Spaces of 𝒮-bounded
multiplier convergent series, Acta Math. Hungar. 87
(2000), no. 1-2, 135–146. MR 1755883
(2001e:46029), http://dx.doi.org/10.1023/A:1006781218759
- 2.
A.
Aizpuru, A.
Gutiérrez-Dávila, and F.
J. Pérez-Fernández, Boolean algebras and uniform
convergence of series, J. Math. Anal. Appl. 284
(2003), no. 1, 89–96. MR 1996119
(2004c:46020), http://dx.doi.org/10.1016/S0022-247X(03)00241-5
- 3.
A. Aizpuru and C. Pérez-Eslava, Matrix methods in summability and weakly unconditionally Cauchy series, Preprint.
- 4.
C.
Bessaga and A.
Pełczyński, On bases and unconditional convergence of
series in Banach spaces, Studia Math. 17 (1958),
151–164. MR 0115069
(22 #5872)
- 5.
Johann
Boos, Classical and modern methods in summability, Oxford
Mathematical Monographs, Oxford University Press, Oxford, 2000. Assisted by
Peter Cass; Oxford Science Publications. MR 1817226
(2002b:40001)
- 6.
Qingying
Bu and Congxin
Wu, Unconditionally convergent series of operators on Banach
spaces, J. Math. Anal. Appl. 207 (1997), no. 2,
291–299. MR 1438915
(98c:47001), http://dx.doi.org/10.1006/jmaa.1997.5218
- 7.
Joseph
Diestel, Sequences and series in Banach spaces, Graduate Texts
in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004
(85i:46020)
- 8.
Joe
Diestel, Hans
Jarchow, and Andrew
Tonge, Absolutely summing operators, Cambridge Studies in
Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge,
1995. MR
1342297 (96i:46001)
- 9.
Richard
Haydon, A nonreflexive Grothendieck space that does not contain
𝑙_{∞}, Israel J. Math. 40 (1981),
no. 1, 65–73. MR 636907
(83a:46028), http://dx.doi.org/10.1007/BF02761818
- 10.
Walter
Schachermayer, On some classical measure-theoretic theorems for
non-sigma-complete Boolean algebras, Dissertationes Math. (Rozprawy
Mat.) 214 (1982), 33. MR 673286
(84d:28015)
- 11.
Charles
Swartz, The Schur lemma for bounded multiplier convergent
series, Math. Ann. 263 (1983), no. 3,
283–288. MR
704294 (84h:46015), http://dx.doi.org/10.1007/BF01457131
- 1.
- A. Aizpuru and F.J. Pérez-Fernández, Spaces of
 -bounded multiplier convergent series, Acta Math. Hungar. 87 1-2 (2000) 135-146. MR 1755883 (2001e:46029)
- 2.
- A. Aizpuru, A. Gutierrez-Dávila, and F.J. Pérez-Fernández, Boolean algebras and uniform convergence of series, J. Math. Anal. Appl. 284 (2003) 89-96. MR 1996119 (2004c:46020)
- 3.
- A. Aizpuru and C. Pérez-Eslava, Matrix methods in summability and weakly unconditionally Cauchy series, Preprint.
- 4.
- C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958) 151-164. MR 0115069 (22:5872)
- 5.
- J. Boos, Classical and Modern Methods in Summability, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. MR 1817226 (2002b:40001)
- 6.
- Q. Bu and C. Wu, Unconditionally convergent series of operators on Banach spaces, J. Math. Anal. Appl. 207 (1997) 291-299. MR 1438915 (98c:47001)
- 7.
- J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92, New York, Springer-Verlag, 1984. MR 737004 (85i:46020)
- 8.
- J. Diestel, H Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics 43, Cambridge University Press, Cambridge, 1995. MR 1342297 (96i:46001)
- 9.
- R. Haydon, A nonreflexive Grothendieck space that does not contain
 , Israel J. Math. 40 (1981) 65-73. MR 636907 (83a:46028)
- 10.
- W. Schachermayer, On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissertationes Math. (Rozprawy Mat.) 214 (1982) 33 pp. MR 673286 (84d:28015)
- 11.
- C. Swartz, The Schur lemma for bounded multiplier convergent series, Math. Ann. 263 3 (1983) 283-288. MR 704294 (84h:46015)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
46B15,
46B25,
46B20
Retrieve articles in all journals
with MSC (2000):
46B15,
46B25,
46B20
Additional Information
Antonio Aizpuru
Affiliation:
Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, Cádiz, 11510, Spain
Email:
antonio.aizpuru@uca.es
Francisco J. García-Pacheco
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, Ohio, 44242
Email:
fgarcia@math.kent.edu
Consuelo Pérez-Eslava
Affiliation:
Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, Cádiz, 11510, Spain
Email:
consuelo.perezeslava@alum.uca.es
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08882-X
PII:
S 0002-9939(07)08882-X
Keywords:
Uniform convergence,
unconditionally convergent series,
weakly unconditionally Cauchy series,
matrix summability,
regular matrices
Received by editor(s):
January 17, 2006
Received by editor(s) in revised form:
August 9, 2006
Posted:
June 21, 2007
Communicated by:
N. Tomczak-Jaegermann
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
|
