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Complemented subspaces of homogeneous polynomials

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Let \(\mathcal {P}_{K} (^{n}E; F)\) (resp. \(\mathcal {P}_{w} (^{n}E; F)\)) denote the subspace of all \(P\in \mathcal {P}(^{n}E; F)\) which are compact (resp. weakly continuous on bounded sets). We show that if \(\mathcal {P}_{K} (^{n}E; F)\) contains an isomorphic copy of \(c_{0}\), then \(\mathcal {P}_{K} (^{n}E; F)\) is not complemented in \(\mathcal {P}(^{n}E; F)\). Likewise we show that if \(\mathcal {P}_{w} (^{n}E; F)\) contains an isomorphic copy of \(c_{0}\), then \(\mathcal {P}_{w}(^{n}E; F)\) is not complemented in \(\mathcal {P}(^{n}E; F)\).

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  1. Aron, R.M.,Diestel, J., Rajappa, A.K.: Weakly continuous functions on Banach spaces containing \(\ell _{1}\). In: Kalton, N., Saab, E. (eds.) Banach Spaces. Lectures Notes in Mathematics, vol. 1166, pp. 1–3. Springer, Berlin (1985)

    Chapter  Google Scholar 

  2. Aron, R.M., Hervés, C., Valdivia, M.: Weakly continuous mappings on Banach spaces. J. Funct. Anal. 52, 189–204 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aron, R.M., Prolla, J.B.: Polynomial approximation of differentiable functions on Banach spaces. J. Reine Angew. Math. 313, 195–216 (1980)

    MathSciNet  MATH  Google Scholar 

  4. Aron, R.M., Schottenloher, M.: Compact holomorphic mappings on Banach spaces and the approximation property. J. Funct. Anal. 21, 730 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bator, E., Lewis, P.: Complemented spaces of operators. Bull. Pol. Acad. Sci. Math. 50(4), 413–416 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Bessaga, C., Pelczynski, A.: On bases and unconditional convergence of series in Banach spaces. Stud. Math. 17, 151–164 (1958)

    MathSciNet  MATH  Google Scholar 

  7. Blasco, F.: Complementation of symmetric tensor products and polynomials. Stud. Math. 123, 165–173 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Çaliskan, E., Rueda, P.: On distinguished polynomials and their projections. Ann. Acad. Sci. Fenn. Math. 37, 595–603 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Diestel, J.: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, vol. 92. Springer, New York (1984)

  10. Diestel, J., Uhl, J.: Vector Measures, Mathematical Surveys Number 15. American Mathematical Society, Providence (1977)

    Book  MATH  Google Scholar 

  11. Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics. Springer, London (1999)

  12. Emmanuele, G.: Remarks on the uncomplemented subspace \({\cal{W}}(E;F)\). J. Funct. Anal. 99, 125–130 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Emmanuele, G.: A remark on the containment of \(c_{0}\) in spaces of compact operators. Math. Proc. Camb. Philos. Soc. 111, 331–335 (1992)

    Article  MATH  Google Scholar 

  14. Feder, M.: On the non-existance of a projection onto the space of compact operators. Can. Math. Bull. 25, 78–81 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghenciu, I.: Complemented spaces of operators. Proc. Am. Math. Soc. 133(9), 2621–2623 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghenciu, I., Lewis, P.: Unconditional convergence in the strong operator topology and \(\ell _{\infty }\). Glasg. Math. J. 53, 583–598 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. González, M.G., Gutiérrez, J.M.: The polynomial property (V). Arch. Math. (Basel) 75(4), 299–306 (2000)

  18. John, K.: On the uncomplemented subspace \({\cal{K}}(X;Y)\). Czechoslov. Math. J. 42, 167–173 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Kalton, N.: Spaces of compact operators. Math. Ann. 208, 267–278 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mujica, J.: Complex Analysis in Banach Spaces. Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions. North-Holland Mathematics Studies 120. Notas de Matemática 107. North-Holland, Amsterdam (1986)

    Google Scholar 

  21. Pelczynski, A.: On Banach spaces containing \(L_{1}(\mu )\). Stud. Math. 30, 231–246 (1968)

    Article  MATH  Google Scholar 

  22. Ryan, R.: Applications of topological tensor products to infinite dimensional holomorphy, Ph.D. thesis, Trinity College, Dublin (1980)

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Correspondence to Sergio A. Pérez.

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Dedicated to the memory of Jorge Mujica (1946–2017)

Sergio A. Pérez was supported by CAPES and CNPq, Brazil.

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Pérez, S.A. Complemented subspaces of homogeneous polynomials. Rev Mat Complut 31, 153–161 (2018).

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