Abstract
A standard technique in infinite dimensional holomorphy, which produced several useful results, uses the Borel transform to represent linear functionals on certain spaces of multilinear operators between Banach spaces as multilinear operators. In this paper, we develop a technique to represent linear functionals, as linear operators, on spaces of multilinear operators that are beyond the scope of the standard technique. Concrete applications to some well-studied classes of multilinear operators, including the class of compact multilinear operators, and to one new class are provided. We can see, in particular, that sometimes our representations hold under conditions less restrictive than those of the related classical ones.
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References
Alencar, R.: An application of Singer’s theorem to homogeneous polynomials, Banach spaces (Mérida, 1992), Contemp. Math., 144, Amer. Math. Soc., Providence, RI, 1–8 (1993)
Botelho, G., Mujica, X.: Spaces of \( (p)\)-nuclear linear and multilinear operators on Banach spaces and their duals. Linear Algebra Appl. 519, 219–237 (2017)
Botelho, G., Mujica, X.: Ideal extensions of classes of linear operators. Stud. Math. 247(3), 285–297 (2019)
Botelho, G., Pellegrino, D., Rueda, P.: On composition ideals of multilinear mappings and homogeneous polynomials. Publ. Res. Inst. Math. Sci. 43(4), 1139–1155 (2007)
Botelho, G., Polac, L.: A polynomial Hutton theorem with applications. J. Math. Anal. Appl. 415(1), 294–301 (2014)
Botelho, G., Torres, E.R.: Hyper-ideals of multilinear operators. Linear Algebra Appl. 482, 1–20 (2015)
Boyd, C., Brown, A.: Duality in spaces of polynomials of degree at most \(n\). J. Math. Anal. Appl. 429, 1271–1290 (2015)
Carando, D., Dimant, V.: Duality in spaces of nuclear and integral polynomials. J. Math. Anal. Appl. 241, 107–121 (2000)
Carothers, N.L.: A Short Course on Banach Space Theory, London Mathematical Society Student Texts 64. Cambridge University Press, Cambridge (2005)
Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland Publishing, North-Holland (1993)
Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)
Dineen, S.: Complex Analysis in Locally Convex Spaces. North-Holland, Amsterdam (1981)
Dineen, S.: Complex Analysis on Infinite Dimensional Spaces, Springer Monographs in Mathematics. Springer, London (1999)
Dwyer, T.A.W.: Convolution equations for vector-valued entire functions of nuclear bounded type. Trans. Am. Math. Soc. 217, 105–119 (1976)
Fávaro, V.V., Jatobá, A.M.: Hypercyclicity, existence and approximation results for convolution operators on spaces of entire functions. Bull. Belg. Math. Soc. Simon Stevin 26(5), 699–723 (2019)
Fávaro, V.V., Mujica, J.: Hypercyclic convolution operators on spaces of entire functions. J. Oper. Theory. 76, 141–158 (2016)
Fávaro, V.V., Mujica, J.: Convolution operators on spaces of entire functions. Math. Nachr. 291, 41–54 (2018)
Fávaro, V.V., Pellegrino, D.: Duality results in Banach and quasi-Banach spaces of homogeneous polynomials and applications. Stud. Math. 240(2), 123–145 (2018)
Floret, K., García, D.: On ideals of polynomials and multilinear mappings between Banach spaces. Arch. Math. (Basel) 81(3), 300–308 (2003)
Floret, K., Hunfeld, S.: Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces. Proc. Am. Math. Soc. 130(5), 1425–1435 (2002)
Gupta, C. P.: Convolution operators and holomorphic mappings on a Banach space, Séminaire d’Analyse Moderne, No. 2, Dept. Math., Université de Sherbrooke, Québec (1969)
Gupta, C. P.: On the Malgrange theorem for nuclearly entire functions of bounded type on a Banach space. Nederl. Akad. Wetensch. Proc. Ser. A73 = Indag. Math. 32, 356–358 (1970)
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II: Function Spaces. Springer, Berlin (1979)
Matos, M.C.: On multilinear mappings of nuclear type. Rev. Mat. Univ. Complut. Madrid 6(1), 61–81 (1993)
Mujica, J.: Linearization of bounded holomorphic mappings on Banach spaces. Trans. Am. Math. Soc. 324, 867–887 (1991)
Mujica, J.: Complex Analysis in Banach Spaces. Dover Publ, New York (2010)
Muro, S., Pinasco, D., Savransky, M.: Strongly mixing convolution operators on Fréchet spaces of holomorphic functions. Integr. Equ. Oper. Theory 80(4), 453–468 (2014)
Pietsch, A.: Operator Ideals. North-Holland Publications, North-Holland (1980)
Pietsch, A.: Ideals of multilinear functionals. In: Proceedings of the second international conference on operator algebras, ideals, and their applications in theoretical physics (Leipzig, 1983). vol. 67, pp. 185–199 (1983)
Ryan, R. A.: Applications of topological tensor products to infinite dimensional holomorphy, Thesis–Trinity College Dublin (1980)
Ryan, R.A.: Introduction to Tensor Products of Banach Spaces. Springer, New York (2002)
Velanga, T.: Ideals of polynomials between Banach spaces revisited. Linear Multilinear Algebra 66(11), 2328–2348 (2018)
Acknowledgements
The authors thank Ariosvaldo M. Jatobá and Ewerton R. Torres for helpful conversations on this subject and the referees for their suggestions and corrections that improved the final presentation of the paper. G. Botelho: Supported by CNPq Grant 304262/2018-8 and Fapemig Grant PPM-00450-17. R. Wood: Supported by a CAPES scholarship.
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Communicated by Juan Seoane Sepúlveda.
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Botelho, G., Wood, R. On the representation of linear functionals on hyper-ideals of multilinear operators. Banach J. Math. Anal. 15, 25 (2021). https://doi.org/10.1007/s43037-020-00108-4
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DOI: https://doi.org/10.1007/s43037-020-00108-4