Holomorphy types and spaces of entire functions of bounded type on Banach spaces. (English) Zbl 1224.46087

Summary: Spaces of entire functions of \(\Theta \)-holomorphy type of bounded type are introduced and results involving these spaces are proved. In particular, we “construct an algorithm” to obtain a duality result via the Borel transform and to prove existence and approximation results for convolution equations. The results we prove generalize previous results of this type due to B. Malgrange [Ann. Inst. Fourier 6, 271–355 (1955/56; Zbl 0071.09002)], C. P. Gupta [Convolution operators and holomorphic mappings on a Banach space. Seminaire d’analyse moderne. No.2. Sherbrooke, Quebec, Canada: Departement de Mathematiques, Universite de Sherbrooke (1969; Zbl 0243.47016)], M. Matos [Absolutely Summing Mappings, Nuclear Mappings and Convolution Equations, IMECC-UNICAMP, State University of Campinas, São Paulo, Brasil (2007), http://www.ime.unicamp.br/rel_pesq/2007/rp03-07.html] and X. Mujica [Aplicações \(\tau (p;q)\)-somantes e \(\sigma (p)\)-nucleares, Thesis, State University of Campinas, São Paulo, Brasil (2006)].


46G20 Infinite-dimensional holomorphy
46G25 (Spaces of) multilinear mappings, polynomials
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[1] S. Banach: Théorie des opérations linéaires. Hafner, New York, 1932.
[2] S. Dineen: Holomorphy types on a Banach space. Stud. Math. 39 (1971), 241–288. · Zbl 0235.32013
[3] V.V. Fávaro: The Fourier-Borel transform between spaces of entire functions of a given type and order. Port. Math. 65 (2008), 285–309. · Zbl 1152.46033
[4] V.V. Fávaro: Convolution equations on spaces of quasi-nuclear functions of a given type and order. Preprint.
[5] K. Floret: Natural norms on symmetric tensor products of normed spaces. Note Mat. 17 (1997), 153–188. · Zbl 0961.46013
[6] C. Gupta: Convolution Operators and Holomorphic Mappings on a Banach Space. Séminaire d’Analyse Moderne, 2. Université de Sherbrooke, Sherbrooke, 1969.
[7] J. Horváth: Topological Vector Spaces and Distribuitions. Addison-Wesley, Reading, 1966.
[8] B. Malgrange: Existence et approximation des équations aux dérivées partielles et des équations des convolutions. Annales de l’Institute Fourier (Grenoble) VI (1955/56), 271–355. · Zbl 0071.09002
[9] A. Martineau: Équations différentielles d’ordre infini. Bull. Soc. Math. Fr. 95 (1967), 109–154. (In French.) · Zbl 0167.44202
[10] M.C. Matos: On the Fourier-Borel transformation and spaces of entire functions in a normed space. In: Functional Analysis, Holomorphy and Approximation Theory II. North-Holland Math. Studies. (G. I. Zapata, ed.). North-Holland, Amsterdam, 1984, pp. 139–170.
[11] M.C. Matos: On convolution operators in spaces of entire functions of a given type and order. In: Complex Analysis, Functional Analysis and Approximation Theory (J. Mujica, ed.). North-Holland, Amsterdam, 1986, pp. 129–171.
[12] M.C. Matos: Absolutely Summing Mappings, Nuclear Mappings and Convolution Equations. IMECC-UNICAMP, 2007, http://www.ime.unicamp.br/rel_pesq/2007/rp03-07.html .
[13] X. Mujica: Aplicações {\(\tau\)} (p; q)-somantes {\(\sigma\)}(p)-nucleares. Thesis. Universidade Estadual de Campinas, 2006.
[14] L. Nachbin: Topology on Spaces of Holomorphic Mappings. Springer, New York, 1969. · Zbl 0172.39902
[15] A. Pietsch: Ideals of multilinear functionals. In: Proc. 2nd Int. Conf. Operator Algebras, Ideals and Their Applications in Theoretical Physics, Leipzin 1983. Teubner, Leipzig, 1984, pp. 185–199. · Zbl 0561.47037
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