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Strictly singular multiplication operators on ℒ(X)

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Abstract

Exploiting several ℓp-factorization results for strictly singular operators, we study the strict singularity of the multiplication operator LARB: T → ATB on ℒ(X) for various Banach spaces X.

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References

  1. F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Graduate Texts in Mathematics, Vol. 233, Springer, New York, 2006.

    MATH  Google Scholar 

  2. C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, 2006.

    Book  Google Scholar 

  3. R. E. Curto and M. Mathieu (eds.), Elementary Operators and their Applications, Operator Theory: Advances and Applications, Vol. 212, Birkhäuser, Basel, 2011.

    Google Scholar 

  4. W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński, Factoring weakly compact operators, Journal of Functional Analysis 17 (1974) 311–327.

    Article  MathSciNet  Google Scholar 

  5. J. Flores, F. L. Hernández, E. M. Semenov and P. Tradacete, Strictly singular and power-compact operators on Banach lattices, Israel Journal of Mathematics 188 (2012) 323–352.

    Article  MathSciNet  Google Scholar 

  6. F. L. Hernández, E. M. Semenov and P. Tradacete, Strictly singular operators on Lp spaces and interpolation, Proceedings of the American Mathematical Society 138 (2010) 675–686.

    Article  MathSciNet  Google Scholar 

  7. J. R. Holub, On subspaces of separable norm ideals, Bulletin of the American Mathematical Society 79 (1973) 446–448.

    Article  MathSciNet  Google Scholar 

  8. W. B. Johnson, Operators into Lp which factor through ℓp, Journal of the London Mathematical Society 14 (1976) 333–339.

    Article  MathSciNet  Google Scholar 

  9. W. B. Johnson and G. Schechtman, Multiplication operators on L(Lp) and ℓp-strictly singular operators, Journal of the European Mathematical Society 10 (2008) 1105–1119.

    Article  Google Scholar 

  10. M. I. Kadec and A. Pelczynski, Bases, lacunary sequences and complemented subspaces in the spaces Lp, Studia Mathematica 21 (1961/62), 161–176.

    Article  MathSciNet  Google Scholar 

  11. N. J. Kalton, Spaces of compact operators, Mathematische Annalen 208 (1974) 267–278.

    Article  MathSciNet  Google Scholar 

  12. T. Kato, Perturbation theory for nullity deficiency and order quantities of linear operators, Journal d’Analyse Mathématique 6 (1958) 273–322.

    Article  Google Scholar 

  13. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I. Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92, Springer, Heidelberg, 1977.

    MATH  Google Scholar 

  14. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II. Function Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 97, Springer, Heidelberg, 1979.

    Book  Google Scholar 

  15. M. Lindström, E. Saksman and H. O. Tylli, Strictly singular and cosingular multiplications, Canadian Journal of Mathematics 57 (2005) 1249–1278.

    Article  MathSciNet  Google Scholar 

  16. M. Lindstroöm and G. Schluüchtermann, Composition of operator ideals, Mathematica Scandinavica 84 (1999) 284–296.

    Article  MathSciNet  Google Scholar 

  17. M. Mathieu (ed.), Elementary Operators and Applications, World Scientific, Singapore, 1992.

    Google Scholar 

  18. V. D. Milman, Operators of class C0and \(C_0^*\), Teorija Funkciĭ, Funkcional’nyĭ Analiz i ih Priloženija 10 (1970) 15–26.

    Google Scholar 

  19. T. Oikhberg and E. Spinu, Ideals of operators on C*-algebras and their preduals, Bulletin of the London Mathematical Society 47 (2015) 156–170.

    Article  MathSciNet  Google Scholar 

  20. A. Pełczyński, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C(S)-spaces, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 13 (1965) 31–36.

    MathSciNet  MATH  Google Scholar 

  21. E. Saksman and H.-O. Tylli, Weak compactness of multiplication operators on spaces of bounded linear operators, Mathematica Scandinavica 70 (1992) 91–111.

    Article  MathSciNet  Google Scholar 

  22. E. Saksman and H.-O. Tylli, Multiplications and elementary operators in the Banach space setting, in Methods in Banach Space Theory, London Mathematical Society Lecture Note Series, Vol. 337, Cambridge University Press, Cambridge, 2006, pp. 253–292.

    Chapter  Google Scholar 

  23. B. Sari, T. Schlumprecht, N. Tomczak-Jaegermann and V. G. Troitsky, On norm closed ideals in L(ℓp,ℓq), Studia Mathemaatica 179 (2007) 239–262.

    Article  MathSciNet  Google Scholar 

  24. L. Weis, On perturbations of Fredholm operators in Lp(μ)-spaces, Proceedings of the American Mathematical Society 67 (1977) 287–292.

    MathSciNet  Google Scholar 

  25. P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, Vol. 25, Cambridge University Press, Cambridge, 1991.

    Book  Google Scholar 

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Authors and Affiliations

  1. Mathematical Sciences Research Centre, Queen’s University Belfast, University Road, Belfast, BT7 1NN, Northern Ireland

    Martin Mathieu

  2. Consejo Superior de Investigaciones Científicas, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera, 13-15, Campus de Cantoblanco UAM, E-28049, Madrid, Spain

    Pedro Tradacete

Authors
  1. Martin Mathieu
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  2. Pedro Tradacete
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Corresponding author

Correspondence to Pedro Tradacete.

Additional information

The second-named author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness through grants MTM2016-75196-P (AEI/FEDER, UE), MTM2016-76808-P (AEI/FEDER, UE), and the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). Support of an LMS Research in Pairs grant is also gratefully acknowledged. We thank Prof. W. B. Johnson for helpful discussions concerning this paper. We are grateful to the referee for carefully reading the paper and pointing out an oversight in a first version of the proof of Theorem 5.5.

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Mathieu, M., Tradacete, P. Strictly singular multiplication operators on ℒ(X). Isr. J. Math. 236, 685–709 (2020). https://doi.org/10.1007/s11856-020-1985-0

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  • DOI: https://doi.org/10.1007/s11856-020-1985-0

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