Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Integral operators in Banach spaces


Author: J. R. Holub
Journal: Proc. Amer. Math. Soc. 29 (1971), 75-80
MSC: Primary 47B05; Secondary 47G05
DOI: https://doi.org/10.1090/S0002-9939-1971-0293445-6
MathSciNet review: 0293445
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Giving a new proof of a result of Grothendieck, it is shown that a weak operator limit of a bounded net of integral maps is again integral and that every integral map is a weak operator limit of a net of finite dimensional maps which are uniformly bounded in nuclear norm. It is also shown that the subnuclear maps defined by Ruckle are precisely the integral maps. Using this result some questions of Ruckle are answered.


References [Enhancements On Off] (What's this?)

  • [1] M. M. Day, Normed linear spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 21, Springer-Verlag, Berlin, 1958. MR 20 #1187. MR 0094675 (20:1187)
  • [2] N. Dunford and J. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [3] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955). MR 17, 763. MR 0075539 (17:763c)
  • [4] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $ {L_p}$-spaces and their applications, Studia Math. 29 (1968), 275-326. MR 37 #6743. MR 0231188 (37:6743)
  • [5] A. Pietsch, Nukleare lokalkonvexe Räume, Schriftenreihe bei der Deutschen Akad. Wissenschaften Berlin. Reihe A, Reine Mathematik, Heft 1, Akademie-Verlag, Berlin, 1965. MR 31 #6114. MR 0181888 (31:6114)
  • [6] W. Ruckle, Decompositions of operator spaces, Rev. Roumaine Math. Pures Appl. 15 (1970), 119-134. MR 0264379 (41:8975)
  • [7] H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [8] R. Schatten, A theory of cross spaces, Ann. of Math. Studies, no. 26, Princeton Univ. Press, Princeton, N. J., 1950. MR 12, 186. MR 0036935 (12:186e)
  • [9] L. Schwartz, Séminaire Schwartz Année 1953/54, Secrétariat mathématique, Paris, 1954. MR 17, 764.
  • [10] C. P. Stegall and J. R. Retherford, Fully nuclear and completely nuclear operators with applications to $ \mathcal{L}{_1}$ and $ \mathcal{L}{_infty}$ spaces, Trans. Amer. Math. Soc. (to appear). MR 0415277 (54:3368)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B05, 47G05

Retrieve articles in all journals with MSC: 47B05, 47G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0293445-6
Keywords: Integral operator, subnuclear operator, nuclear operator, weak operator topology
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society