Geodesic
AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES
Glasgow mathematical journal, Tome 56 (2014) no. 2, pp. 427-437

Voir la notice de l'article provenant de la source Cambridge University Press

Let 1 ≤ p < ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.
DOI : 10.1017/S0017089513000360
Mots-clés : Primary 46B20, Secondary 46B28, 46B50 
KARN, ANIL KUMAR; SINHA, DEBA PRASAD. AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES. Glasgow mathematical journal, Tome 56 (2014) no. 2, pp. 427-437. doi: 10.1017/S0017089513000360
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[1] 1.Bartle, N., Dunford, N. and Schwartz, J., Weak compactness and vector measures}, Canad. J. Math. 7 (1955), 289–305. Google Scholar | DOI

[2] 2.Bourgain, J. and Diestel, J., Limited operators and strictly cosingularity}, Math. Nachr. 119 (1984), 55–58. Google Scholar | DOI

[3] 3.Brace, B., Transformation on Banach spaces, PhD Dissertation (Cornell University Ithaca, NY, 1953). Google Scholar

[4] 4.Castillo, J. M. F. and Sánchez, F., Dunford–Pettis properties of continuous vector valued function spaces}, Rev. Mat. Univ. Comput. Madrid 6 (1993), 43–59. Google Scholar

[5] 5.Castillo, J. M. F. and Sánchez, F., Weakly p-compact, p-Banach–Saks and super reflexive Banach spaces}, J. Math. Anal. Appl. 185 (1994), 256–261. Google Scholar

[6] 6.Choi, Y. S. and Kim, J. M., The dual space of and the p-approximation property (pre-print). Google Scholar

[7] 7.Delgado, J. M., Oja, E., Pineiro, C. and Serrano, E., The p-approximation property in terms of density of finite rank operators}, J. Math. Anal. Appl. 354 (2009), 159–164. Google Scholar | DOI

[8] 8.Delgado, J. M., Pineiro, C. and Serrano, E., Operators whose adjoints are quasi p-nuclear}, Studia Math. 197 (3) (2010), 291–304. Google Scholar

[9] 9.Diestel, J., A survey of results related to the Dunford–Pettis property}, Contemp. Math. 2 (1980), 15–60. Google Scholar | DOI

[10] 10.Diestel, J., Jarchow, H. and Tonge, A., Absolutely summing operators (Cambridge University Press, Cambridge, UK, 1995). Google Scholar

[11] 11.Dunford, N. and Pettis, B. J., Linear operations on summable functions}, Trans. Amer. Math. Soc. 47 (1940), 323–392. Google Scholar

[12] 12.Gelfand, I. M., Abstrakte Funktionen und lineare operatoren}, Rev. Roumaine Math. Pures Appl. 5 (1938), 742–752. Google Scholar

[13] 13.Grothendieck, A., Sur les applications linéares faiblement compactes d'espaces du type C(K), Canad. J. Math. 5 (1953), 129–173. Google Scholar | DOI

[14] 14.Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires}, {Mem. Amer. Math. Soc.}, Vol. 16 (AMS, Providence, RI, 1955). Google Scholar

[15] 15.Kwapień, S.Some remarks on (p, q)-absolutely summing operators in l-spaces, Studia Math. 29 (1968), 327–337. Google Scholar | DOI

[16] 16.Lindenstruss, J. and Pelczynski, A., Absolutely summing operators in -spaces and their applications}, Studia Math. 29 (1968), 275–326. Google Scholar

[17] 17.Phillips, R., On linear transformations}, Trans. Amer. Math. Soc. 48 (1940), 516–541. Google Scholar | DOI

[18] 18.Pietsch, A., Absolute p-summierende Abbildungen in normierten Räumen}, Studia Math. 28 (1967), 333–353. Google Scholar | DOI

[19] 19.Pineiro, C. and Delgado, J. M., p-convergent sequences and Banach spaces in which p-compact sets are q-compact}, Proc. Amer. Math. Soc. 139 (3) (2011), 957–967. Google Scholar | DOI

[20] 20.Rosenthal, H. P., On subspaces of L, Ann. Math. 97 (2) (1973), 344–373. Google Scholar

[21] 21.Saphar, P. D., Produits tensoriels d'espaces de Banach et classes d'applications linéaires}, Studia Math. 38 (1970), 71–100. Google Scholar

[22] 22.Sinha, D. P. and Karn, A. K., Compact operators whose adjoints factor through subspaces of l}, Studia Math. 150 (2002), 17–33. Google Scholar

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