Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem

[EN] Let (X, Sigma, mu) be a finite measure space and consider a Banach function space Y(mu). Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Kothe Bochner (vectorvalued) function spaces. We analyze operator-valued kernels Gamma that define integration maps L-Gamma between Kothe-Bochner spaces of Hilbert-valued functions Y(mu; kappa). We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces L-P(gd mu; kappa) and L-P (hd mu; kappa) - where 1/p + 1/p' = 1 - under the assumption of p-concavity of Y(mu). Equivalently, a new kernel obtained by multiplying Gamma by scalar functions can be given in such a way that the kernel operator is defined from L-P (mu; kappa) to L-P (mu; kappa) in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels.The second author acknowledges the support of the Ministerio de Economia y Competitividad (Spain), under project MTM2014-53009-P (Spain). The third author acknowledges the support of the Ministerio de Ciencia, Innovacion y Universidades (Spain), Agencia Estatal de Investigacion, and FEDER under project MTM2016-77054-C2-1-P (Spain).Agud Albesa, L.; Calabuig, JM.; Sánchez Pérez, EA. (2020). Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem. Analysis and Applications. 18(3):359-383. https://doi.org/10.1142/S0219530519500179S359383183Agud, L., Calabuig, J. M., & Sánchez Pérez, E. A. (2011). The weak topology on q-convex Banach function spaces. Mathematische Nachrichten, 285(2-3), 136-149. doi:10.1002/mana.201000030CARMELI, C., DE VITO, E., & TOIGO, A. (2006). VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES OF INTEGRABLE FUNCTIONS AND MERCER THEOREM. Analysis and Applications, 04(04), 377-408. doi:10.1142/s0219530506000838CARMELI, C., DE VITO, E., TOIGO, A., & UMANITÀ, V. (2010). VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES AND UNIVERSALITY. Analysis and Applications, 08(01), 19-61. doi:10.1142/s0219530510001503Cerdà, J., Hudzik, H., & Mastyło, M. (1996). Geometric properties of Köthe–Bochner spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 120(3), 521-533. doi:10.1017/s0305004100075058Chavan, S., Podder, S., & Trivedi, S. (2018). Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces. Journal of Mathematical Analysis and Applications, 466(2), 1337-1358. doi:10.1016/j.jmaa.2018.06.062Christmann, A., Dumpert, F., & Xiang, D.-H. (2016). On extension theorems and their connection to universal consistency in machine learning. Analysis and Applications, 14(06), 795-808. doi:10.1142/s0219530516400029Defant, A. (2001). Positivity, 5(2), 153-175. doi:10.1023/a:1011466509838Defant, A., & Sánchez Pérez, E. A. (2004). Maurey–Rosenthal factorization of positive operators and convexity. Journal of Mathematical Analysis and Applications, 297(2), 771-790. doi:10.1016/j.jmaa.2004.04.047De Vito, E., Umanità, V., & Villa, S. (2013). An extension of Mercer theorem to matrix-valued measurable kernels. Applied and Computational Harmonic Analysis, 34(3), 339-351. doi:10.1016/j.acha.2012.06.001Eigel, M., & Sturm, K. (2017). Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization. Optimization Methods and Software, 33(2), 268-296. doi:10.1080/10556788.2017.1314471Fasshauer, G. E., Hickernell, F. J., & Ye, Q. (2015). Solving support vector machines in reproducing kernel Banach spaces with positive definite functions. Applied and Computational Harmonic Analysis, 38(1), 115-139. doi:10.1016/j.acha.2014.03.007Galdames Bravo, O. (2014). Generalized Kӧthe $p$-dual spaces. Bulletin of the Belgian Mathematical Society - Simon Stevin, 21(2). doi:10.36045/bbms/1400592625Lin, P.-K. (2004). Köthe-Bochner Function Spaces. doi:10.1007/978-0-8176-8188-3Lindenstrauss, J., & Tzafriri, L. (1979). Classical Banach Spaces II. doi:10.1007/978-3-662-35347-9Meyer-Nieberg, P. (1991). Banach Lattices. Universitext. doi:10.1007/978-3-642-76724-1Okada, S., Ricker, W. J., & Sánchez Pérez, E. A. (2008). Optimal Domain and Integral Extension of Operators. doi:10.1007/978-3-7643-8648-1Zhang, H., & Zhang, J. (2013). Vector-valued reproducing kernel Banach spaces with applications to multi-task learning. Journal of Complexity, 29(2), 195-215. doi:10.1016/j.jco.2012.09.00

Banach Lattice Structures and Concavifications in Banach Spaces

[EN] Let (Omega,sigma,mu) be a finite measure space and consider a Banach function space Y(mu). We say that a Banach space E is representable by Y(mu) if there is a continuous bijection I:Y(mu)-> E. In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.The authors would like to thank the referees for their valuable comments, which helped to improve the manuscript. The work of the second author was supported by the Ministerio de Ciencia e Innovacion, Agencia Estatal del Investigacion (Spain) and FEDER under project #PGC2018-095366-B-100. The work of the fourth author was supported by the Ministerio de Ciencia e Innovacion, Agencia Estatal del Investigacion (Spain) and FEDER under project #MTM2016 77054-C2-1-P. We did not receive any funds for covering the costs of publishing in open access.Agud Albesa, L.; Calabuig, JM.; Juan, MA.; Sánchez Pérez, EA. (2020). Banach Lattice Structures and Concavifications in Banach Spaces. Mathematics. 8(1):1-20. https://doi.org/10.3390/math8010127S12081Bu, Q., Buskes, G., Popov, A. I., Tcaciuc, A., & Troitsky, V. G. (2012). The 2-concavification of a Banach lattice equals the diagonal of the Fremlin tensor square. Positivity, 17(2), 283-298. doi:10.1007/s11117-012-0166-8Buskes, G., & van Rooij, A. (2001). Squares of Riesz Spaces. Rocky Mountain Journal of Mathematics, 31(1). doi:10.1216/rmjm/1008959667Troitsky, V. G., & Zabeti, O. (2013). Fremlin tensor products of concavifications of Banach lattices. Positivity, 18(1), 191-200. doi:10.1007/s11117-013-0239-3Defant, A. (2001). Positivity, 5(2), 153-175. doi:10.1023/a:1011466509838Defant, A., & Sánchez Pérez, E. A. (2004). Maurey–Rosenthal factorization of positive operators and convexity. Journal of Mathematical Analysis and Applications, 297(2), 771-790. doi:10.1016/j.jmaa.2004.04.047Berezhnoĭ, E. I., & Maligranda, L. (2005). Representation of Banach Ideal Spaces and Factorization of Operators. Canadian Journal of Mathematics, 57(5), 897-940. doi:10.4153/cjm-2005-035-0Reisner, S. (1981). A factorization theorem in Banach lattices and its application to Lorentz spaces. Annales de l’institut Fourier, 31(1), 239-255. doi:10.5802/aif.825Curbera, G. P. (1992). Operators intoL 1 of a vector measure and applications to Banach lattices. Mathematische Annalen, 293(1), 317-330. doi:10.1007/bf01444717Curbera, G. P., Okada, S., & Ricker, W. J. (2019). Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces. Quaestiones Mathematicae, 43(5-6), 783-812. doi:10.2989/16073606.2019.1605423Delgado, O., & Soria, J. (2007). Optimal domain for the Hardy operator. Journal of Functional Analysis, 244(1), 119-133. doi:10.1016/j.jfa.2006.12.011Bravo, O. G. (2016). On the Optimal Domain of the Laplace Transform. Bulletin of the Malaysian Mathematical Sciences Society, 40(1), 389-408. doi:10.1007/s40840-016-0402-7Mockenhaupt, G., & Ricker, W. J. (2008). Optimal extension of the Hausdorff-Young inequality. Journal für die reine und angewandte Mathematik (Crelles Journal), 2008(620). doi:10.1515/crelle.2008.054Leśnik, K., & Tomaszewski, J. (2017). Pointwise mutipliers of Orlicz function spaces and factorization. Positivity, 21(4), 1563-1573. doi:10.1007/s11117-017-0485-xDe Jager, P., & Labuschagne, L. E. (2019). Multiplication operators on non-commutative spaces. Journal of Mathematical Analysis and Applications, 475(1), 874-894. doi:10.1016/j.jmaa.2019.03.001Bartle, R. G., Dunford, N., & Schwartz, J. (1955). Weak Compactness and Vector Measures. Canadian Journal of Mathematics, 7, 289-305. doi:10.4153/cjm-1955-032-1Calabuig, J. M., Delgado, O., & Sánchez Pérez, E. A. (2008). Generalized perfect spaces. Indagationes Mathematicae, 19(3), 359-378. doi:10.1016/s0019-3577(09)00008-1Maligranda, L., & Persson, L. E. (1989). Generalized duality of some Banach function spaces. Indagationes Mathematicae (Proceedings), 92(3), 323-338. doi:10.1016/s1385-7258(89)80007-1Schep, A. R. (2009). Products and factors of Banach function spaces. Positivity, 14(2), 301-319. doi:10.1007/s11117-009-0019-2Delgado, O., & Sánchez Pérez, E. A. (2010). Summability Properties for Multiplication Operators on Banach Function Spaces. Integral Equations and Operator Theory, 66(2), 197-214. doi:10.1007/s00020-010-1741-7Sánchez Pérez, E. A. (2015). Product spaces generated by bilinear maps and duality. Czechoslovak Mathematical Journal, 65(3), 801-817. doi:10.1007/s10587-015-0209-yGillespie, T. A. (1981). Factorization in Banach function spaces. Indagationes Mathematicae (Proceedings), 84(3), 287-300. doi:10.1016/1385-7258(81)90040-8Calderón, A. (1964). Intermediate spaces and interpolation, the complex method. Studia Mathematica, 24(2), 113-190. doi:10.4064/sm-24-2-113-190Calabuig, J. M., Delgado, O., & Sánchez Pérez, E. A. (2010). Factorizing operators on Banach function spaces through spaces of multiplication operators. Journal of Mathematical Analysis and Applications, 364(1), 88-103. doi:10.1016/j.jmaa.2009.10.034Calabuig, J. M., Gregori, P., & Sánchez Pérez, E. A. (2008). Radon–Nikodým derivatives for vector measures belonging to Köthe function spaces. Journal of Mathematical Analysis and Applications, 348(1), 469-479. doi:10.1016/j.jmaa.2008.07.024Bu, Q., & Buskes, G. (2012). Polynomials on Banach lattices and positive tensor products. Journal of Mathematical Analysis and Applications, 388(2), 845-862. doi:10.1016/j.jmaa.2011.10.001Kusraeva, Z. A. (2018). Powers of quasi-Banach lattices and orthogonally additive polynomials. Journal of Mathematical Analysis and Applications, 458(1), 767-780. doi:10.1016/j.jmaa.2017.09.01

On the smoothness of L p of a positive vector measure

The final publication is available at Springer via http://dx.doi.org/10.1007/s00605-014-0666-7We investigate natural sufficient conditions for a space L p(m) of pintegrable functions with respect to a positive vector measure to be smooth. Under some assumptions on the representation of the dual space of such a space, we prove that this is the case for instance if the Banach space where the vector measure takes its values is smooth. We give also some examples and show some applications of our results for determining norm attaining elements for operators between two spaces L p(m1) and Lq (m2) of positive vector measures m1 and m2.Professor Agud and professor Sanchez-Perez authors gratefully acknowledge the support of the Ministerio de Economia y Competitividad (Spain), under project #MTM2012-36740-c02-02. Professor Calabuig gratefully acknowledges the support of the Ministerio de Economia y Competitividad (Spain), under project #MTM2011-23164.Agud Albesa, L.; Calabuig Rodriguez, JM.; Sánchez Pérez, EA. (2015). On the smoothness of L p of a positive vector measure. Monatshefte für Mathematik. 178(3):329-343. https://doi.org/10.1007/s00605-014-0666-7S3293431783Beauzamy, B.: Introduction to Banach Spaces and Their Geometry. North-Holland, Amsterdam (1982)Diestel, J., Uhl, J.J.: Vector measures. In: Mathematical Surveys, vol. 15. AMS, Providence (1977)Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., Sánchez-Pérez, E.A.: Spaces of p-integrable functions with respect to a vector measure. Positivity 10, 1–16 (2006)Ferrando, I., Rodríguez, J.: The weak topology on $$L^p$$ L p of a vector measure. Topol. Appl. 155(13), 1439–1444 (2008)Godefroy, G.: Boundaries of a convex set and interpolation sets. Math. Ann. 277(2), 173–184 (1987)Howard, R., Schep, A.R.: Norms of positive operators on $$L^p$$ L p -spaces. Proc. Am. Math. Soc. 109(1), 135–146 (1990)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1977)Meyer-Nieberg, P.: Banach Latticces. Universitext, Springer-Verlag, Berlin (1991)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)Schep, A.: Products and factors of Banach function spaces. Positivity 14(2), 301–319 (2010

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

Characterization of methicillin-resistant Staphylococcus aureus strains colonizing the nostrils of Spanish children

Objective: To characterize the Staphylococcus aureus strains colonizing healthy Spanish children. Methods: Between March and July 2018, 1876 Spanish children younger than 14 years attending primary healthcare centers were recruited from rural and urban areas. Staphylococcus aureus colonization of the anterior nostrils was analyzed. MecA and mecC genes, antibiotic susceptibility, and genotyping according to the spa were determined in all strains, and the following toxins were examined: Panton-Valentine leucocidin (pvl), toxic shock syndrome toxin (tst), and exfoliative toxins (eta, etb, etd). Multilocus sequence typing (MLST) and staphylococcal cassette chromosome (SCCmec) typing were performed on methicillin-resistant Staphylococcus aureus (MRSA) strains, as well as pulsed-field gel electrophoresis (PFGE). Results: 619 strains were isolated in 1876 children (33%), and 92% of them were sent for characterization to the Spanish National Centre of Microbiology (n = 572). Twenty (3.5%) of these strains were mecA-positive. Several spa types were detected among MRSA, being t002 the most frequently observed (30%), associating with SCCmec IVc. Among MSSA, 33% were positive for tst, while only 0.73% were positive for pvl. The 20 MRSA strains were negative for pvl, and 6 (30%) harbored the tst gene. Conclusions: methicillin-resistant Staphylococcus aureus nasal colonization in Spanish children is rare, with t002 being the most observed spa type, associated with SCCmec IVc. None of the MRSA strains produced pvl, but up to 30% of S. aureus strains were positive for tst