Multiplication Semigroups on Banach Function Spaces

In this paper we characterize multiplication operators induced by operator valued maps on Banach function spaces. We also study multiplication semigroups and stability of these operators.Comment: We want to withdraw the paper due to the paper needs a careful revision and some proper citations are to be provided. Also there are certain gaps in the results which need to be taken car

Monotonicity Properties of Musielak–Orlicz Spaces and Dominated Best Approximation in Banach Lattices

AbstractCriteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and uniform monotonicity of a Musielak–Orlicz space endowed with the Amemiya norm and its subspace of order continuous elements are given in the cases of nonatomic and the counting measure space. To complete the results of Kurc (J. Approx. Theory69(1992), 173–187), criteria for upper local uniform monotonicity of these spaces equipped with the Luxemburg norm are also given. Some applications to dominated best approximation are presented

Differentiability of L-p of a vector measure and applications to the Bishop-Phelps-Bollobas property

[EN] We study the properties of Gâteaux, Fréchet, uniformly Fréchet and uniformly Gâteaux smoothness of the space Lp(m) of scalar p-integrable functions with respect to a positive vector measure m with values in a Banach lattice. Applications in the setting of the Bishop-Phelps-Bollobás property (both for operators and bilinear forms) are also given.Research supported by Ministerio de Economia y Competitividad and FEDER under projects MTM2012-36740-c02-02 (L. Agud and E.A. Sanchez-Perez), MTM201453009-P (J.M. Calabuig) and MTM2014-54182-P (S. Lajara). S. Lajara was also supported by project 19275/PI/14 funded by Fundacion Seneca-Agencia de Ciencia y Tecnologia de la Region de Murcia within the framework of PCTIRM 2011-2014.Agud Albesa, L.; Calabuig, JM.; Lajara, S.; Sánchez Pérez, EA. (2017). Differentiability of L-p of a vector measure and applications to the Bishop-Phelps-Bollobas property. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 111(3):735-751. https://doi.org/10.1007/s13398-016-0327-xS7357511113Acosta, M.D., Aron, R.M., García, D., Maestre, M.: The Bishop–Phelps–Bollobás theorem for operators. J. Funct. Anal. 254(11), 2780–2799 (2008)Acosta, M.D., Becerra-Guerrero, J., Choi, Y.S., García, D., Kim, S.K., Lee, H.J., Maestre, M.: The Bishop–Phelps–Bollobás theorem for bilinear forms and polinomials. J. Math. Soc. Jpn 66(3), 957–979 (2014)Acosta, M.D., Becerra-Guerrero, J., García, D., Maestre, M.: The Bishop–Phelps–Bollobás theorem for bilinear forms. Trans. Am. Math. Soc. 11, 5911–5932 (2013)Agud, L., Calabuig, J.M., Sánchez Pérez, E.A.: On the smoothness of $$L^p$$ L p of a positive vector measure. Monatsh. Math. 178(3), 329–343 (2015)Aron, R.M., Cascales, B., Kozhushkina, O.: The Bishop–Phelps–Bollobás theorem and Asplund operators. Proc. Am. Math. Soc. 139, 3553–3560 (2011)Bishop, E., Phelps, R.R.: A proof that every Banach space is subreflexive. Bull. Am. Math. Soc. 67, 97–98 (1961)Bollobás, B.: An extension to the theorem of Bishop and Phelps. Bull. Lond. Math. Soc. 2, 181–182 (1970)Cascales, B., Guirao, A.J., Kadets, V.: A Bishop–Phelps–Bollobás theorem type theorem for uniform algebras. Adv. Math. 240, 370–382 (2013)Choi, Y.S., Song, H.G.: The Bishop–Phelps–Bollobás theorem fails for bilinear forms on $$\ell _1\times \ell _1$$ ℓ 1 × ℓ 1 . J. Math. Anal. Appl. 360, 752–753 (2009)Deville, R., Godefroy, G., Zizler, V.: Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Appl. Math., vol. 64, Longman, Harlow (1993)Diestel, J., Uhl, J.J.: Vector Measures. Math. Surveys, vol. 15, AMS, Providence, RI (1977)Fabian, M., Godefroy, G., Montesinos, V., Zizler, V.: Inner characterizations of weakly compactly generated Banach spaces and their relatives. J. Math. Anal. Appl. 297, 419–455 (2004)Fabian, M., Godefroy, G., Zizler, V.: The structure of uniformly Gâteaux smooth Banach spaces. Israel J. Math. 124, 243–252 (2001)Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory: The Basis for Linear and Nonlinear Analysis. CMS Books in Mathematics, Springer, New York (2011)Fabian, M., Lajara, S.: Smooth renormings of the Lebesgue–Bochner function space $$L^1(\mu, X)$$ L 1 ( μ , X ) . Stud. Math. 209(3), 247–265 (2012)Ferrando, I., Rodríguez, J.: The weak topology on $$L^p$$ L p of a vector measure. Top. Appl. 155(13), 1439–1444 (2008)Hájek, P., Johanis, M.: Smooth analysis in Banach spaces. De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter (2014)Kim, S.K.: The Bishop–Phelps–Bollobás theorem for operators from $$c_0$$ c 0 to uniformly convex spaces. Israel J. Math. 197, 425–435 (2013)Kim, S.K., Lee, H.J.: The Bishop–Phelps–Bollobás theorem for operators from $$C(K)$$ C ( K ) to uniformly convex spaces. J. Math. Anal. Appl. 421(1), 51–58 (2015)Hudzik, H., Kamińska, A., Mastylo, M.: Monotonocity and rotundity properties in Banach lattices. Rock. Mount J. Math. 30(3), 933–950 (2000)Kutzarova, D., Troyanski, S.L.: On equivalent norms which are uniformly convex or uniformly differentiable in every direction in symmetric function spaces. Serdica 11, 121–134 (1985)Okada, S., Ricker, W.J., Sánchez-Pérez, E.A.: Optimal Domain and Integral Extension of Operators Acting in Function Spaces. Operator Theory: Advances and Applications, vol. 180. Birkhäuser Verlag, Basel (2008