Universal families for conull FK spaces

Snyder, A. K.

doi:10.1090/S0002-9947-1984-0742431-1

Universal families for conull FK spaces

Author: A. K. Snyder
Journal: Trans. Amer. Math. Soc. 284 (1984), 389-399
MSC: Primary 46A45; Secondary 40H05
DOI: https://doi.org/10.1090/S0002-9947-1984-0742431-1
MathSciNet review: 742431
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper considers the problem of determining a useful family of sequence spaces which is universal for conull $\text {FK}$ spaces in the following sense: An $\text {FK}$ space is conull if and only if it contains a member of the family. In the equivalent context of weak wedge spaces, an appropriate family of subspaces of boundedness domains ${m_A}$ of matrices is shown to be universal. Most useful is the fact that the members of this family exhibit unconditional sectional convergence. The latter phenomenon is known for wedge spaces. Another family of spaces which is universal for conull spaces among semiconservative spaces is provided. The spaces are designed to simplify gliding humps arguments. Improvements are thereby obtained for some pseudoconull type theorems of Bennett and Kalton. Finally, it is shown that conull spaces must contain pseudoconull $\text {BK}$ algebras.

G. Bennett and N. J. Kalton, $FK$-spaces containing $c_{0}$, Duke Math. J. 39 (1972), 561–582. MR 310597
G. Bennett and N. J. Kalton, Addendum to: “$FK$-spaces containing $c_{0}$”, Duke Math. J. 39 (1972), 819–821. MR 313758
G. Bennett, The gliding humps technique for $FK$-spaces, Trans. Amer. Math. Soc. 166 (1972), 285–292. MR 296564, DOI https://doi.org/10.1090/S0002-9947-1972-0296564-9
G. Bennett, A new class of sequence spaces with applications in summability theory, J. Reine Angew. Math. 266 (1974), 49–75. MR 344846, DOI https://doi.org/10.1515/crll.1974.266.49
J. Copping, Inclusion theorems for conservative summation methods, Nederl. Akad. Wetensch. Proc. Ser. A 61 = Indag. Math. 20 (1958), 485–499. MR 0099552

R. Devos, Distinguished subsets and matrix maps between $FK$ spaces, Ph. D. dissertation, Lehigh University, 1971.

D. J. H. Garling, On topological sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967), 997–1019. MR 218880, DOI https://doi.org/10.1017/s0305004100042031
Haskell P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from $L^{p}\,(\mu )$ to $L^{r}\,(\nu )$, J. Functional Analysis 4 (1969), 176–214. MR 0250036, DOI https://doi.org/10.1016/0022-1236%2869%2990011-1
John J. Sember, Variational $FK$ spaces and two-norm convergence, Math. Z. 119 (1971), 153–159. MR 280908, DOI https://doi.org/10.1007/BF01109968
A. K. Snyder, Conull and coregular ${\rm FK}$ spaces, Math. Z. 90 (1965), 376–381. MR 185315, DOI https://doi.org/10.1007/BF01112357
A. K. Snyder, Consistency theory in semiconservative spaces, Studia Math. 71 (1981/82), no. 1, 1–13. MR 651321, DOI https://doi.org/10.4064/sm-71-1-1-13
A. K. Snyder and A. Wilansky, Inclusion theorems and semiconservative $FK$ spaces, Rocky Mountain J. Math. 2 (1972), no. 4, 595–603. MR 310496, DOI https://doi.org/10.1216/RMJ-1972-2-4-595
Albert Wilansky, Summability through functional analysis, North-Holland Mathematics Studies, vol. 85, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 91. MR 738632
Karl Zeller, Faktorfolgen bei Limitierungsverfahren, Math. Z. 56 (1952), 134–151 (German). MR 49342, DOI https://doi.org/10.1007/BF01175031