Bounded holomorphic functions attaining their norms in the bidual

Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain their norms, is dense in $\mathcal{A}_u(X)$. The result holds also for functions with values in a dual space or in a Banach space with the so-called property $(\beta)$. For this, we establish first a Lindenstrauss type theorem for continuous polynomials. We also present some counterexamples for the Bishop-Phelps theorem in the analytic and polynomial cases where our results apply.Comment: Accepted in Publ. Res. Inst. Math. Sc

Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

The final publication is available at Springer via http://dx.doi.org/10.1007/s00013-012-0458-zWe investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H∞. Applications for composition operators on weighted Bloch spaces are given. © 2012 Springer Basel.1. The authors are thankful to the referee for pointing to us the references [15] and [16] and their relevance in the present research. 2. The research of Bonet was partially supported by MICINN and FEDER Project MTM2010-15200 and by GV project Prometeo/2008/101 and project ACOMP/2012/090.Bonet Solves, JA.; Lindström, M.; Wolf, E. (2012). Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions. Archiv der Mathematik. 99(6):537-546. https://doi.org/10.1007/s00013-012-0458-zS537546996Bierstedt K.D., Bonet J., Galbis A.: Weighted spaces of holomorphic functions on bounded domains. Michigan Math. J. 40, 271–297 (1993)Bierstedt K.D., Bonet J., Taskinen J.: Associated weights and spaces of holomorphic functions. Studia Math. 127, 137–168 (1998)J. Bonet, P. Domański, and M. Lindström, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Canad, Math. Bull. 42, no. 2, (1999), 139–148Bonet J. et al.: Composition operators between weighted Banach spaces of analytic functions. J. Austral. Math. Soc. Ser. A 64, 101–118 (1998)Bonet J., Lindström M, Wolf E.: Isometric weighted composition operators on weighted Banach spaces of type H ∞. Proc. Amer. Math. Soc. 136, 4267–4273 (2008)Bonet J, Wolf E.: A note on weighted spaces of holomorphic functions. Archiv Math. 81, 650–654 (2003)Contreras M.D, Hernández-Díaz A.G.: Weighted composition operators in weighted banach spaces of analytic functions. J. Austral. Math. Soc. Ser. A 69, 41–60 (2000)Cowen C., MacCluer B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)J. Diestel, Geometry of Banach Spaces. Selected Topics, Lecture Notes in Math. vol. 485, Springer, Berlin, 1975.Hammond C.: On the norm of a composition operator with linear fractional symbol. Acta Sci. Math. (Szeged) 69, 813–829 (2003)Hosokawa T., Izuchi K., Zheng D.: Isolated points and essential components of composition operators on H ∞. Proc. Amer. Math. Soc. 130, 1765–1773 (2001)Hosokava T., Ohno S.: Topological strusctures of the sets of composition operatorson the Bloch spaces. J. Math. anal. Appl. 303, 499–508 (2005)Lusky W.: On the isomorphy classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–45 (2006)Martín M.: Norm-attaining composition operators on the Bloch spaces. J. Math. Anal. Appl. 369, 15–21 (2010)A. Montes-Rodríguez, The Pick-Schwarz lemma and composition operators on Bloch spaces, International Workshop on Operator Theory (Cefalu, 1997), Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 167–170.Montes-Rodríguez A.: The essential norm of a composition operator on Bloch spaces. Pacific J. Math. 188, 339–351 (1999)Montes-Rodríguez A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. London Math. Soc. 61, 872–884 (2000)J.H. Shapiro, Composition Operators and Classical Function Theory, Springer, 1993.K. Zhu, Operator Theory in Function Spaces, Second Edition. Amer. Math. Soc., 2007

Banach spaces with polynomial numerical index 1

We characterize Banach spaces with polynomial numerical index 1 when they have the Radon-Nikod\'ym property. The holomorphic numerical index is introduced and the characterization of the Banach space with holomorphic numerical index 1 is obtained when it has the Radon-Nikod\'ym property

A Lindenstrauss theorem for some classes of multilinear mappings

Under some natural hypotheses, we show that if a multilinear mapping belongs to some Banach multlinear ideal, then it can be approximated by multilinear mappings belonging to the same ideal whose Arens extensions simultaneously attain their norms. We also consider the class of symmetric multilinear mappings.Comment: 11 page

A characterisation of C*-algebras through positivity of functionals

We show that a unital involutive Banach algebra, with identity of norm one and continuous involution, is a C*-algebra, with the given involution and norm, if every continuous linear functional attaining its norm at the identity is positive.Comment: 3 pages. Some typos corrected. Final version, to appear in Ann. Funct. Ana