• Badea, Gabriela (Department of Mathematics, Ovidius University of Constanta) ;
  • Popa, Dumitru (Department of Mathematics, Ovidius University of Constanta)
  • Received : 2016.05.04
  • Published : 2017.05.01


We give the necessary condition for an operator defined on a cartesian product of $c_0(\mathcal{X})$ spaces to be summing or dominated and we show that for the multiplication operators this condition is also sufficient. By using these results, we show that ${\Pi}_s(c_0,{\ldots},c_0;c_0)$ contains a copy of $l_s(l^m_2{\mid}m{\in}\mathbb{N})$ for s > 2 or a copy of $1_s(l^m_1{\mid}{\in}\mathbb{N})$, for any $l{\leq}S$ < ${\infty}$. Also ${\Delta}_{s_1,{\ldots},s_n}(c_0,{\ldots},c_0;c_0)$ contains a copy of $l_{{\upsilon}_n(s_1,{\ldots},s_n)}$ if ${\upsilon}_n(s_1,{\ldots},s_n){\leq}2$ or a copy of $l_{{\upsilon}_n(s_1,{\ldots},s_n)}(l^m_2{\mid}m{\in}\mathbb{N})$ if 2 < ${\upsilon}_n(s_1,{\ldots},s_n)$, where ${\frac{1}{{\upsilon}_n(s_1,{\ldots},s_n})}={\frac{1}{s_1}}+{\cdots}+{\frac{1}{s_n}}$. We find also the necessary and sufficient conditions for bilinear operators induced by some method of summability to be 1-summing or 2-dominated.


Supported by : Romanian National Authority for Scientific Research


  1. G. Badea, On nuclear and multiple summing bilinear operators on $c_0{\times}c_0$, Quaest Math. 33 (2010), no. 2, 253-261.
  2. G. Badea and D. Popa, Swartz type results for nuclear and multiple 1-summing bilinear operators on $c_0(X){\times}c_0(Y)$, Positivity 19 (2015), no. 3, 475-487.
  3. G. Botelho, Cotype and absolutely summing multilinear mappings and homogeneous polynomials, Proc. Royal Irish Acad. Sect. A 97 (1997), no. 2, 145-153.
  4. R. Cilia and J. M. Gutierrez, Nuclear and integral polynomials, J. Aust. Math. Soc. 76 (2004), no. 2, 269-280.
  5. A. Defant and K. Floret, Tensor norms and operator ideals, Math. Studies 176, North-Holland, 1993.
  6. J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge University Press, 1995.
  7. S. Dineen, Complex analysis in locally convex spaces, North Holland Math Studies, vol 57, North Holland, Amsterdam 1981.
  8. N. Dunford and J. T. Schwartz, Linear Operators I: General Theory, Pure and Applied Mathematics, vol. 7, Interscience, New York, 1958.
  9. K. Floret and D. Garcia, On ideals of polynomials and multilinear mappings between Banach spaces, Arch. Math. (Basel) 81 (2003), no. 3, 300-308.
  10. A. Nahoum, Applications radonifiantes dans l'espace des series convergentes. II: Les resultats, Sem. Maurey-Schwartz 1972-1973, exp XXV (1973),
  11. A. Pietsch, Ideals of multilinear functionals, Proc. 2nd Int. Conf. Operator Alg., Leibzig, Teubner Texte Math. 62 (1983), 185-199.
  12. A. Pietsch, Eigenvalues and s-numbers, Cambridge Stud. Adv. Math. 13, Cambridge University Press, 1987.
  13. D. Popa, Examples of operators on C[0, 1] distinguishing certain operator ideals, Arch. Math. 88 (2007), no. 4, 349-357.
  14. D. Popa, Averages and compact, absolutely summing and nuclear operators on C(${\Omega}$), J. Korean Math. Soc. 47 (2010), no. 5, 899-924.
  15. N. Tomczak-Jagermann, Banach-Mazur distances and finite dimensional operator ideals, Pitman Monographs, vol. 38, Harlow: Longman Scientific & Technical, 1989.
  16. P. Wojtaszczyk, Banach spaces for analysts, Cambridge Stud. Adv. Math. 25, Cambridge University Press, 1996.

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