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Monatshefte für Mathematik

, Volume 172, Issue 3–4, pp 259–275 | Cite as

Trace formulas for nuclear operators in spaces of Bochner integrable functions

  • Julio DelgadoEmail author
Article

Abstract

The paper is devoted to trace formulas for nuclear operators in spaces of Bochner integrable functions. We characterise nuclearity for integral operators on such spaces and develop a trace formula for general kernels applying vector-valued maximal functions.

Keywords

Integral operators Nuclear operators Bochner integral Vector-valued maximal function Trace formula 

Mathematics Subject Classification (2010)

Primary 47B10; Secondary 47G10 47B38 60G46 46E40 

Notes

Acknowledgments

This work has been supported by a Marie Curie International Incoming Fellowship of the European Commision 7th Framework Programme under contract number 301599. I would also like to thank an anonymous refeere for the valuable comments on the results and presentation helping to improve this manuscript.

References

  1. 1.
    Brislawn, C.: Kernels of trace class operators. Proc. Am. Math. Soc. 104, 1181–1190 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brislawn, C.: Traceable integral Kernels on countably generated measure spaces. Pac. J. Math. 150(2), 229–240 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Castro, M.H., Menegatto, V.A., Peron, A.P.: Traceability of positive integral operators in the absence of a metric. Banach J. Math. Anal. 6(2), 98–112 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chatterji, S.D.: Martingales of Banach valued random variable. Bull. Am. Math. Soc. 66, 395–398 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chatterji, S.D.: A note on the convergence of Banach-space valued martingales. Math. Ann. 153(2), 142–149 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Defant, A., Floret, K.: Tensor norms and opertor ideals. North-Holland Mathematics Studies vol. 176. North-Holland, Amsterdam (1993)Google Scholar
  7. 7.
    Delgado, J.: A trace formula for nuclear operators on \(L^p\). Pseudo-differential operators: complex analysis and partial differential equations. In: Schulze, B.-W., Wong, M.W. (eds.) Operator Theory, Advances and Applications, vol. 205, pp. 181–193. Birkhäuser, Basel (2009)Google Scholar
  8. 8.
    Delgado, J.: The trace of nuclear operators on \(L^p(\mu )\) for \(\sigma \)-finite Borel measures on second second countable spaces. Integral Equ. Oper. Theory 68, 61–74 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Delgado, J., Wong, M.W.: \(L^{p}\)-nuclear pseudodifferential operators on \({\mathbb{Z}}\) and \({\mathbb{S}}\). Proc. Amer. Math. Soc. (in press)Google Scholar
  10. 10.
    Doob, J.L.: Stochastic Processes. Wiley, New York (1953)zbMATHGoogle Scholar
  11. 11.
    Doob, J.L.: Measure theory, Graduate Texts in Mathematics, 143. Springer, New York (1994)Google Scholar
  12. 12.
    Fefferman, Ch., Stein, E.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gohberg, I., Goldberg, S., Krupnik, N.: Traces and determinants of linear operators. Integral Equ. Oper. Theory 26, 136–187 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gohberg, I., Goldberg, S., Krupnik, N.: Traces and determinants of linear operators. Birkhäuser, Basel (2001)Google Scholar
  15. 15.
    Grothendieck, A.: Produits tensoriels topologiques et espaces nuclaires. Mem. Am. Math. Soc. 16, 140 (1955)MathSciNetGoogle Scholar
  16. 16.
    Hytönen, T., McIntosh, A., Portal, P.: Kato’s square root problem in Banach spaces. J. Funct. Anal. 254(3), 675–726 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mansuy, R.: Histoire de Martingales, \(43^{e}\) anne. Math. Sci. Hum. 169, 105–113 (2005)MathSciNetGoogle Scholar
  18. 18.
    Mansuy, R.: The origins of the word “martingale”. Translated from the French [MR2135182] by Ronald Sverdlove. J. Électron. Hist. Probab. Stat. 5(1), p. 10 (2009)Google Scholar
  19. 19.
    Pietsch, A.: Operator Ideals. North-Holland, Amsterdam (1980)zbMATHGoogle Scholar
  20. 20.
    Pietsch, A.: Eigenvalues and s-numbers. Cambridge University Press, New York (1986)Google Scholar
  21. 21.
    Poincaré, H.: Sur les determinants d’ordre infini. Bull. S.M.F. 14, 77–90 (1886)zbMATHGoogle Scholar
  22. 22.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  23. 23.
    Stein, E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  24. 24.
    Weidman, J.: Integraloperatoren der spurklasse. Math. Ann. 163, 340–345 (1966)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK

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