Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 453–471 | Cite as

Measures of weak non-compactness in spaces of nuclear operators

  • Jan Hamhalter
  • Ondřej F. K. KalendaEmail author


We show that in the space of nuclear operators from \(\ell ^q(\Lambda )\) to \(\ell ^p(J)\) (where \(p,q\in (1,\infty )\)) the two natural ways of measuring weak non-compactness coincide. We also provide explicit formulas for these measures. As a consequence the same is proved for preduals of atomic von Neumann algebras.


Measure of weak non-compactness Space of nuclear operators Space of compact operators Predual of an atomic von Neumann algebra 

Mathematics Subject Classification

46B04 46B50 46B28 46L10 47B10 



  1. 1.
    Akemann, C.A., Anderson, J.: Lyapunov theorems for operator algebras. Mem. Am. Math. Soc. 94(458), iv+88 (1991)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Angosto, C., Cascales, B.: The quantitative difference between countable compactness and compactness. J. Math. Anal. Appl. 343(1), 479–491 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Angosto, C., Cascales, B.: Measures of weak noncompactness in Banach spaces. Topol. Appl. 156(7), 1412–1421 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Astala, K., Tylli, H.-O.: Seminorms related to weak compactness and to Tauberian operators. Math. Proc. Camb. Philos. Soc. 107(2), 367–375 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bendová, H., Kalenda, O.F.K., Spurný, J.: Quantification of the Banach–Saks property. J. Funct. Anal. 268(7), 1733–1754 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blackadar, B.: Operator Algebras, Theory of $C^*$-algebras and von Neumann Algebras, Operator Algebras and Non-commutative Geometry, III, Encyclopaedia of Mathematical Sciences, vol. 122. Springer, Berlin (2006)Google Scholar
  7. 7.
    Cascales, B., Marciszewski, W., Raja, M.: Distance to spaces of continuous functions. Topol. Appl. 153(13), 2303–2319 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cascales, B., Kalenda, O.F.K., Spurný, J.: A quantitative version of James’s compactness theorem. Proc. Edinb. Math. Soc. (2) 55(2), 369–386 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    De Blasi, F .S.: On a property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 21(69)(3—-4), 259–262 (1977)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fabian, M., Hájek, P., Montesinos, V., Zizler, V.: A quantitative version of Krein’s theorem. Rev. Mat. Iberoam. 21(1), 237–248 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fabian, M., Montesinos, V., Zizler, V.: A characterization of subspaces of weakly compactly generated Banach spaces. J. Lond. Math. Soc. (2) 69(2), 457–464 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Granero, A.S.: An extension of the Krein-Šmulian theorem. Rev. Mat. Iberoam. 22(1), 93–110 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Granero, A.S., Hernández, J.M., Pfitzner, H.: The distance ${\rm dist}({\cal{B}}, X)$ when $\cal{B}$ is a boundary of $B(X^{\ast \ast })$. Proc. Am. Math. Soc. 139(3), 1095–1098 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grothendieck, A.: Espaces vectoriels topologiques. Instituto de Matemática Pura e Aplicada, Universidade de São Paulo, São Paulo (1954)Google Scholar
  16. 16.
    Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc. 16, 140 (1955)zbMATHGoogle Scholar
  17. 17.
    Kačena, M., Kalenda, O.F.K., Spurný, J.: Quantitative Dunford–Pettis property. Adv. Math. 234, 488–527 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras. Vol. II, Graduate Studies in Mathematics, vol. 16, Advanced theory, Corrected reprint of the 1986 original. American Mathematical Society, Providence (1997)Google Scholar
  19. 19.
    Kalenda, O.F.K., Spurný, J.: Quantification of the reciprocal Dunford–Pettis property. Stud. Math. 210(3), 261–278 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Meise, R., Vogt, D.: Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2, Translated from the German by M. S. Ramanujan and revised by the authors. The Clarendon Press, Oxford University Press, New York (1997)Google Scholar
  21. 21.
    Ryan, R.A.: Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Electrical EngineeringCzech Technical University in PraguePrague 6Czech Republic
  2. 2.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

Personalised recommendations