On the local moduli of squareness

[EN] We introdu e the notions of pointwise modulus of squareness and lo al modulus of squareness of a normed spa e X . This answers a question of C. Benítez, K. Przesªawski and D. Yost about the de nition of a sensible lo alization of the modulus of squareness. Geometri al properties of the norm of X (Fré het smoothness, Gâteaux smoothness, lo al uniform onvexity or stri t onvexity) are hara terized in terms of the behaviour of these moduli.The author wishes to express his gratitude to Professor R. Deville for helpful omments and suggestions whi h improved the ontent of this paperGuirao Sánchez, AJ. (2008). On the local moduli of squareness. Studia Mathematica. 184(2):175-189. doi:10.4064/sm184-2-6S175189184

On the moduly of convexity

[EN] It is known that, given a Banach space (X, parallel to center dot parallel to), the modulus of convexity associated to this space delta X is a non-negative function, nondecreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation delta x(epsilon)/epsilon(2) 0 is a constant. We show that, given a function f satisfying these properties then, there exists a Banach space in such a way its modulus of convexity is equivalent to f, in Figiel's sense. Moreover this Banach space can be taken to be two-dimensional.The first author was supported by grants MTM2005-08379 of MECD (Spain), 00690/PI/04 of Fundación Séneca (CARM, Spain), and AP2003-4453 of MECD (Spain).Guirao Sánchez, AJ.; Hajek, P. (2007). On the moduly of convexity. Proceedings of the American Mathematical Society. 135(10):3233-3240. https://doi.org/10.1090/S0002-9939-07-09030-2S323332401351

Schauder bases under uniform renormings

[EN] Let X be a separable superreflexive Banach space with a Schauder basis. We prove the existence of an equivalent uniformly smooth (resp. uniformly rotund) renorming under which the given basis is monotone.First author supported by the grants MTM2005-08379 of MECD (Spain), 00690/PI/04 of Fundación Séneca (CARM, Spain) and AP2003-4453 of MECD (Spain), Second author supported by AV0Z10190503 and A100190502.Guirao Sánchez, AJ.; Hajek, P. (2007). Schauder bases under uniform renormings. Positivity. 11(4):627-638. https://doi.org/10.1007/s11117-007-2067-9S627638114R. Deville, G. Godefroy, V. Zizler, Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys 64, Longman Ed (1993).M. Fabian, P. Habala, P. Hájek, V. Montesinos, J. Pelant, V. Zizler, Functional analysis and infinite dimensional geometry. Canadian Math. Soc. Books, Springer Verlag, (2001).M. Fabian, V. Montesinos, V. Zizler, Smoothness in Banach spaces. Selected problems. Rev. R. Acad. Cien. Serie A Mat. 100, (2006), 101–125.T. Figiel, On the moduli of convexity and smoothness. Studia Math. 56, (1976), 121–155.M. Zippin, A remark on bases and reflexivity in Banach spaces. Isr. J. Math. 6, (1968), 74–79.P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm. Isr. J. Math 13, (1972), 281–288

The Bishop-Phelps-Bollobás property for numerical radius in l(1)(C)

We show that the set of bounded linear operators from X to X admits a Bishop Phelps Bollobas type theorem for numerical radius whenever X is l(1)(C) or c(0)(C). As an essential tool we provide two constructive versions of the classical Bishop-Phelps-Bollobas theorem for l(1)(C).The research of the first named author was supported in part by MICINN and FEDER (project MTM2011-25377), by Fundacion Seneca (project 08848/PI/08), by Generalitat Valenciana (GV/2010/036), and by Universidad Politecnica de Valencia (project PAID-06-09-2829). The research of the second named author is supported by Kent State UniversityGuirao Sánchez, AJ.; Kozhushkina, O. (2013). The Bishop-Phelps-Bollobás property for numerical radius in l(1)(C). Studia Mathematica. 218(1):41-54. https://doi.org/10.4064/sm218-1-3S4154218

On the Bishop-Phelps-Bollobás property for numeri9cal radius in C(K) spaces

[EN] We study the Bishop-Phelps-Bollobás property for numerical radius within the framework of C(K) spaces. We present several sufficient conditions on a compact space K ensuring that C(K) has the Bishop-Phelps-Bollobás property for numerical radius. In particular, we show that C(K) has such property whenever K is metrizable.Research supported by Ministerio de Economía y Competitividad and FEDER under project MTM2011-25377. A. Avilés was supported by Ramón y Cajal contract (RYC-2008-02051). A.J. Guirao was supported by Generalitat Valenciana (GV/2010/036)Avilés López, A.; Guirao Sánchez, AJ.; Rodríguez, J. (2014). On the Bishop-Phelps-Bollobás property for numeri9cal radius in C(K) spaces. Journal of Mathematical Analysis and Applications. 419(1):395-421. doi:10.1016/j.jmaa.2014.04.039S395421419