zbMATH — the first resource for mathematics

Local properties of accessible injective operator ideals. (English) Zbl 0926.46054
In contrast to Pisier’s example of a non-accessible maximal Banach ideal, the author exhibits a large class of maximal Banach ideals which are accessible. The construction is based on a characterization of total accessibility of injective hulls of maximal Banach ideals (if \((\mathcal A,\mathbf A)\) is a maximal Banach ideal, then \((\mathcal A^{inj},\mathbf A^{inj})\) is totally accessible if and only if \((\mathcal A\circ \mathcal A^*,\mathbf A\circ \mathbf A^\ast)\subseteq (\mathcal P_1,\mathbf P_1)\), the ideal of absolutely summing operators) and an operator version of Grothendieck’s inequality. As a consequence, it is obtained, for instance, that if \((\mathcal A,\mathbf A)\) is a maximal Banach ideal such that \((\mathcal D_2,\mathbf D_2)\subseteq (\mathcal A,\mathbf A)\subseteq (\mathcal L_1,\mathbf L_1)\) and both \((\mathcal A,\mathbf A)\) and \((\mathcal A^*, \mathbf A^\ast)\) are metrically \(\varepsilon \)-tensorstable, then \((\mathcal A^{inj}, \mathbf A^{inj})\) is totally accessible. Several other criteria for right- and total accessibility are established, and applications to normed products of operator ideals are also given.

46M05 Tensor products in functional analysis
47L20 Operator ideals
47A80 Tensor products of linear operators
Full Text: DOI arXiv EuDML
[1] B. Carl, A. Defant, and M. S. Ramanujan: On tensor stable operator ideals. Michigan Math. J. 36 (1989), 63-75. · Zbl 0669.47025
[2] A. Defant: Produkte von Tensornormen. Habilitationsschrift. Oldenburg 1986.
[3] A. Defant and K. Floret: Tensor Norms and Operator Ideals. North-Holland Amsterdam, London, New York, Tokio, 1993. · Zbl 0774.46018
[4] J. E. Gilbert and T. Leih: Factorization, tensor products and bilinear forms in Banach space theory. Notes in Banach spaces, Univ. of Texas Press, Austin, 1980, pp. 182-305. · Zbl 0471.46053
[5] Y. Gordon, D. R. Lewis, and J. R Retherford: Banach ideals of operators with applications. J. Funct. Analysis 14 (1973), 85-129. · Zbl 0272.47024
[6] A. Grothendieck: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8 (1956), 1-79. · Zbl 0074.32303
[7] H. Jarchow: Locally convex spaces. Teubner, 1981. · Zbl 0466.46001
[8] H. Jarchow and R. Ott: On trace ideals. Math. Nachr. 108 (1982), 23-37. · Zbl 0523.47030
[9] H. P. Lotz: Grothendieck ideals of operators in Banach spaces. Lecture notes, Univ. Illinois, Urbana, 1973.
[10] J. Lindenstrauss and H. P. Rosenthal: The Lp-spaces. Israel J. Math. 7 (1969), 325-349. · Zbl 0205.12602
[11] F. Oertel: Konjugierte Operatorenideale und das \({\mathcal A}\)-lokale Reflexivitätsprinzip. Dissertation. Kaiserslautern, 1990.
[12] F. Oertel: Operator ideals and the principle of local reflexivity. Acta Universitatis Carolinae-Mathematica et Physica 33 (1992), no. 2, 115-120. · Zbl 0803.47038
[13] A. Pietsch: Operator Ideals. North-Holland Amsterdam, London, New York, Tokio, 1980. · Zbl 0455.47032
[14] A. Pietsch: Eigenvalues and s-numbers. Cambridge Studies in Advanced Mathematics 13 (1987). · Zbl 0615.47019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.