Holomorphic functions with large cluster sets
Thiago R. Alves
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal do Amazonas, 69.077-000, Manaus, Brazil
Search for more papers by this authorCorresponding Author
Daniel Carando
Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, Argentina
IMAS-UBA-CONICET, Argentina
Correspondence
Daniel Carando, Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, Argentina.
Email: [email protected]
Search for more papers by this authorThiago R. Alves
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal do Amazonas, 69.077-000, Manaus, Brazil
Search for more papers by this authorCorresponding Author
Daniel Carando
Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, Argentina
IMAS-UBA-CONICET, Argentina
Correspondence
Daniel Carando, Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, Argentina.
Email: [email protected]
Search for more papers by this authorAbstract
We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed unit ball of the bidual in the infinite dimensional case). We show that this set is strongly -algebrable for all separable Banach spaces. For specific spaces including or duals of Lorentz sequence spaces, we have strongly -algebrability and spaceability even for the subalgebra of uniformly continous holomorphic functions on the ball.
REFERENCES
- 1M. D. Acosta and L. Lourenço, Shilov boundary for holomorphic functions on some classical Banach spaces, Studia Math. 179 (2007), 27–39.
- 2T. R. Alves, Lineability and algebrability of the set of holomorphic functions with a given domain of existence, Studia Math. 220 (2014), 157–167.
- 3T. R. Alves and G. Botelho, Holomorphic functions with distinguished properties on infinite dimensional spaces, Q. J. Math. 70 (2019), 797–811.
- 4R. Aron, L. Bernal-González, D. Pellegrino, and J. B. Seoane-Sepúlveda, Lineability: The search for linearity in mathematics, Monogr. Res. Notes Math., Chapman & Hall/CRC, Boca Raton, FL, 2016.
- 5R. M. Aron, D. Carando, T. W. Gamelin, S. Lassalle, and M. Maestre, Cluster values of analytic functions on a Banach space, Math. Ann. 353 (2012), 293–303.
- 6R. M. Aron, D. Carando, S. Lassalle, and M. Maestre, Cluster values of holomorphic functions of bounded type, Trans. Amer. Math. Soc. 368 (2016), 2355–2369.
- 7R. M. Aron, D. García, and M. Maestre, Linearity in non-linear problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 95 (2001), 7–12.
- 8L. Bernal-González, Vector spaces of non-extendable holomorphic functions, J. Anal. Math. 134 (2018), 769–786.
- 9L. Bernal-González, A. Bonilla, M. Calderón-Moreno, and J. A. Prado-Bassas, Universal Taylor series with maximal cluster sets, Rev. Mat. Iberoam. 25 (2009), no. 2, 757–780.
- 10L. Bernal-González, D. Pellegrino, and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. 51 (2014), 71–130.
- 11S. Charpentier and Ł. Kosiński, Wild boundary behaviour of holomorphic functions in domains of , arXiv:1907.05455 (preprint).
- 12A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), 351–356.
- 13J. Diestel, Sequences and series in banach spaces, Grad. Texts in Math., vol. 92, Springer-Verlag, New York, 1984.
10.1007/978-1-4612-5200-9 Google Scholar
- 14S. Dineen, Complex analysis on infinite dimensional spaces, Springer, 1999.
10.1007/978-1-4471-0869-6 Google Scholar
- 15J. D. Farmer, Fibers over the sphere of a uniformly convex Banach space, Michigan Math. J. 45 (1998), 211–226.
- 16P. Galindo and A. Miralles, Interpolating sequences for bounded analytic functions, Proc. Amer. Math. Soc. 135 (2007), 3225–3231.
- 17D. Garling, On symmetric sequence spaces, Proc. Lond. Math. Soc. 16 (1966), no. 3, 85–106.
10.1112/plms/s3-16.1.85 Google Scholar
- 18V. I. Gurarij, Linear spaces composed of non-differentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), no. 5, 13–16.
- 19W. B. Johnson and S. Ortega Castillo, The cluster value problem in spaces of continuous functions, Proc. Amer. Math. Soc. 143 (2015), no. 4, 1559–1568.
- 20J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I – Sequence spaces, Springer-Verlag, Berlin, 1977.
- 21J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II - Function spaces, Springer-Verlag, Berlin, 1979.
10.1007/978-3-662-35347-9 Google Scholar
- 22J. López-Salazar, Lineability of the set of holomorphic mappings with dense range, Studia Math. 210 (2012), no. 2, 177–188.
- 23J. Mujica, Complex analysis in Banach spaces, Dover Publications, New York, 2010.
- 24S. Ortega Castillo and A. Prieto, The polynomial cluster value problem, J. Math. Anal. Appl. 461 (2018), no. 2, 1459–1470.
- 25W. L. C. Sargent, Some sequence spaces related to the spaces, J. Lond. Math. Soc. 35 (1960), 161–171.
10.1112/jlms/s1-35.2.161 Google Scholar
- 26I. J. Schark, Maximal ideals in an algebra of bounded analytic functions, J. Math. Mech. 10 (1961), 735–746.
