Grothendieck operators on tensor products

Domański, P.; Lindström, M.; Schlüchtermann, G.

doi:10.1090/S0002-9939-97-03763-5

Grothendieck operators on tensor products
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by P. Domański, M. Lindström and G. Schlüchtermann PDF

Proc. Amer. Math. Soc. 125 (1997), 2285-2291 Request permission

Abstract:

We prove that for Banach spaces $E,F,G,H$ and operators $T\in \mathcal {L}(E,G)$, $S\in \mathcal {L}(F,H)$ the tensor product $T\otimes S:E \otimes _\varepsilon F\to G\otimes _\varepsilon H$ is a Grothendieck operator, provided $T$ is a Grothendieck operator and $S$ is compact.

References

Freda E. Alexander, Compact and finite rank operators on subspaces of $l_{p}$, Bull. London Math. Soc. 6 (1974), 341–342. MR 355661, DOI 10.1112/blms/6.3.341
J. Bonet, M. Lindström, and M. Valdivia, Two theorems of Josefson-Nissenzweig type for Fréchet spaces, Proc. Amer. Math. Soc. 117 (1993), no. 2, 363–364. MR 1136233, DOI 10.1090/S0002-9939-1993-1136233-5
Pilar Cembranos, $C(K,\,E)$ contains a complemented copy of $c_{0}$, Proc. Amer. Math. Soc. 91 (1984), no. 4, 556–558. MR 746089, DOI 10.1090/S0002-9939-1984-0746089-2
P. Domański and M. Lindström, Grothendieck spaces and duals of injective tensor products, Bull. London Math. Soc. 28 (1996), 617–626.
J. Diestel and B. Faires, Remarks on the classical Banach operator ideals, Proc. Amer. Math. Soc. 58 (1976), 189–196. MR 454701, DOI 10.1090/S0002-9939-1976-0454701-6
Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
T. Figiel, Factorization of compact operators and applications to the approximation problem, Studia Math. 45 (1973), 191–210. (errata insert). MR 336294, DOI 10.4064/sm-45-2-191-210
Francisco J. Freniche, Barrelledness of the space of vector valued and simple functions, Math. Ann. 267 (1984), no. 4, 479–486. MR 742894, DOI 10.1007/BF01455966
Francisco J. Freniche, Grothendieck locally convex spaces of continuous vector valued functions, Pacific J. Math. 120 (1985), no. 2, 345–355. MR 810776
Kamil John, On the compact nonnuclear operator problem, Math. Ann. 287 (1990), no. 3, 509–514. MR 1060689, DOI 10.1007/BF01446908
William B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337–345. MR 290133, DOI 10.1007/BF02771684
N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267–278. MR 341154, DOI 10.1007/BF01432152
Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR 582655
Frank Räbiger, Beiträge zur Strukturtheorie der Grothendieck-Räume, Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse [Reports of the Heidelberg Academy of Science. Section for Mathematics and Natural Sciences], vol. 85, Springer-Verlag, Berlin, 1985 (German). MR 828457, DOI 10.1007/978-3-642-45612-1
Eero Saksman and Hans-Olav Tylli, Weak compactness of multiplication operators on spaces of bounded linear operators, Math. Scand. 70 (1992), no. 1, 91–111. MR 1174205, DOI 10.7146/math.scand.a-12388
G. Schlüchtermann, Weak compactness in $L_\infty (\mu ,X)$, J. Funct. Anal. 125 (1994), no. 2, 379–388. MR 1297673, DOI 10.1006/jfan.1994.1129