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

Spectra of weighted algebras of holomorphic functions

We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach-Stone type question is addressed.Comment: 25 pages Corrected typo

Extendibility of bilinear forms on banach sequence spaces

[EN] We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize c(0) in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.The second author was supported by MICINN Project MTM2011-22417.DANIEL CARANDO; Sevilla Peris, P. (2014). Extendibility of bilinear forms on banach sequence spaces. Israel Journal of Mathematics. 199(2):941-954. https://doi.org/10.1007/s11856-014-0003-9S9419541992F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Graduate Texts in Mathematics, Vol. 233, Springer, New York, 2006.R. Arens, The adjoint of a bilinear operation, Proceedings of the American Mathematical Society 2 (1951), 839–848.R. Arens, Operations induced in function classes, Monatshefte für Mathematik 55 (1951), 1–19.R. M. Aron and P. D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bulletin de la Société Mathématique de France 106 (1978), 3–24.S. Banach, Sur les fonctionelles linéaires, Studia Mathematica 1 (1929), 211–216.S. Banach, Théorie des opérations linéaires, (Monogr. Mat. 1) Warszawa: Subwncji Funduszu Narodowej. VII, 254 S., Warsaw, 1932.D. Carando, Extendible polynomials on Banach spaces, Journal of Mathematical Analysis and Applications 233 (1999), 359–372.D. Carando, Extendibility of polynomials and analytic functions on l p, Studia Mathematica 145 (2001), 63–73.D. Carando, V. Dimant and P. Sevilla-Peris, Limit orders and multilinear forms on lp spaces, Publications of the Research Institute for Mathematical Sciences 42 (2006), 507–522.J. M. F. Castillo, R. García, A. Defant, D. Pérez-García and J. Suárez, Local complementation and the extension of bilinear mappings, Mathematical Proceedings of the Cambridge Philosophical Society 152 (2012), 153–166.J. M. F. Castillo, R. García and J. A. Jaramillo, Extension of bilinear forms on Banach spaces, Proceedings of the American Mathematical Society 129 (2001), 3647–3656.P. Cembranos and J. Mendoza, The Banach spaces ℓ ∞(c 0) and c 0(ℓ ∞) are not isomorphic, Journal of Mathematical Analysis and Applications 367 (2010), 461–463.A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Mathematics Studies, Vol. 176, North-Holland Publishing Co., Amsterdam, 1993.A. Defant and C. Michels, Norms of tensor product identities, Note di Matematica 25 (2005/06), 129–166.J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press, Cambridge, 1995.D. J. H. Garling, On symmetric sequence spaces, Proceedings of the London Mathematical Society (3) 16 (1966), 85–106.A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79.H. Hahn, Über lineare Gleichungssysteme in linearen Räumen, Journal für die Reine und Angewandte Mathematik 157 (1927), 214–229.R. C. James, Bases and reflexivity of Banach spaces, Annals of Mathematics (2) 52 (1950), 518–527.H. Jarchow, C. Palazuelos, D. Pérez-García and I. Villanueva, Hahn-Banach extension of multilinear forms and summability, Journal of Mathematical Analysis and Applications 336 (2007), 1161–1177.W. B. Johnson and L. Tzafriri, On the local structure of subspaces of Banach lattices, Israel Journal of Mathematics 20 (1975), 292–299.P. Kirwan and R. A. Ryan, Extendibility of homogeneous polynomials on Banach spaces, Proceedings of the American Mathematical Society 126 (1998), 1023–1029.J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in Lp-spaces and their applications, Studia Mathematica 29 (1968), 275–326.J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Vol. 97, Springer-Verlag, Berlin, 1979. Function spaces.G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conference Series in Mathematics, Vol. 60, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986.M. Fernndez-Unzueta and A. Prieto, Extension of polynomials defined on subspaces, Mathematical Proceedings of the Cambridge Philosophical Society 148 (2010), 505–518.W. L. C. Sargent, Some sequence spaces related to the lp spaces, Journal of the London Mathematical Society 35 (1960), 161–171.N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 38, Longman Scientific & Technical, Harlow, 1989

Almost sure-sign convergence of Hardy-type Dirichlet series

[EN] Hartman proved in 1939 that the width of the largest possible strip in the complex plane on which a Dirichlet series is uniformly a.s.- sign convergent (i.e., converges uniformly for almost all sequences of signs epsilon (n) = +/- 1) but does not convergent absolutely, equals 1/2. We study this result from a more modern point of view within the framework of so-called Hardytype Dirichlet series with values in a Banach space.Supported by CONICET-PIP 11220130100329CO, PICT 2015-2299 and UBACyT 20020130100474BA. Supported by MICINN MTM2017-83262-C2-1-P. Supported by MICINN MTM2017-83262-C2-1-P and UPV-SP20120700.Carando, D.; Defant, A.; Sevilla Peris, P. (2018). Almost sure-sign convergence of Hardy-type Dirichlet series. Journal d Analyse Mathématique. 135(1):225-247. https://doi.org/10.1007/s11854-018-0034-yS2252471351A. Aleman, J.-F. Olsen, and E. Saksman, Fourier multipliers for Hardy spaces of Dirichlet series, Int. Math. Res. Not. IMRN 16 (2014), 4368–4378.R. Balasubramanian, B. Calado, and H. Queffélec, The Bohr inequality for ordinary Dirichlet series Studia Math. 175 (2006), 285–304.F. Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136 (2002), 203–236.F. Bayart, A. Defant, L. Frerick, M. Maestre, and P. Sevilla-Peris, Monomial series expansion of Hp-functions and multipliers ofHp-Dirichlet series, Math. Ann. 368 (2017), 837–876.F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda, The Bohr radius of the n-dimensional polydisk is equivalent to $$\sqrt {\left( {\log n} \right)/n}$$ ( log n ) / n , Adv. Math. 264 (2014), 726–746.F. Bayart, H. Queffélec, and K. Seip, Approximation numbers of composition operators on Hp spaces of Dirichlet series, Ann. Inst. Fourier (Grenoble) 66 (2016), 551–588.H. F. Bohnenblust and E. Hille. On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32 (1931), 600–622.H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variablen in der Theorie der Dirichlet–schen Reihen $$\sum {\frac{{{a_n}}}{{{n^s}}}}$$ ∑ a n n s , Nachr. Ges.Wiss. Göttingen, Math. Phys. Kl., 1913, pp. 441–488.D. Carando, A. Defant, and P. Sevilla-Peris, Bohr’s absolute convergence problem for Hp- Dirichlet series in Banach spaces, Anal. PDE 7 (2014), 513–527.D. Carando, A. Defant, and P. Sevilla-Peris, Some polynomial versions of cotype and applications, J. Funct. Anal. 270 (2016), 68–87.B. J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. (3) 53 (1986), 112–142.R. de la Bretèche. Sur l’ordre de grandeur des polynômes de Dirichlet, Acta Arith. 134 (2008), 141–148.A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounäies, and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (2011), 485–497.A. Defant, D. García, M. Maestre, and D. Pérez-García, Bohr’s strip for vector valued Dirichlet series, Math. Ann. 342 (2008), 533–555.A. Defant, M. Maestre, and U. Schwarting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231 (2012), 2837–2857.A. Defant and A. Pérez, Hardy spaces of vector-valued Dirichlet series, StudiaMath. (to appear), 2018 DOI: 10.4064/sm170303-26-7.A. Defant, U. Schwarting, and P. Sevilla-Peris, Estimates for vector valued Dirichlet polynomials, Monatsh. Math. 175 (2014), 89–116.J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995.P. Hartman, On Dirichlet series involving random coefficients, Amer. J. Math. 61 (1939), 955–964.H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L2(0, 1), Duke Math. J. 86 (1997), 1–37.A. Hildebrand, and G. Tenenbaum, Integers without large prime factors, J. Thor. Nombres Bordeaux 5 (1993), 411–484.S. V. Konyagin and H. Queffélec, The translation 1/2 in the theory of Dirichlet series, Real Anal. Exchange 27 (2001/02) 155–175.J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Springer-Verlag, Berlin, 1979.B. Maurizi and H. Queffélec, Some remarks on the algebra of bounded Dirichlet series, J. Fourier Anal. Appl. 16 (2010), 676–692.H. Queffélec, H. Bohr’s vision of ordinary Dirichlet series; old and new results, J. Anal. 3 (1995), 43–60.H. Queffélec and M. Queffélec, Diophantine Approximation and Dirichlet Series, Hindustan Book Agency, New Delhi, 2013.G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995

El rendimiento en Matemática y Física desde la mirada de estudiantes universitarios

En la convicción de que los alumnos deben ser educados como protagonistas del proceso de aprendizaje y enseñanza, se administró una encuesta a un total de 261 estudiantes universitarios de las áreas de Matemática y Física (de distintas Facultades y Universidades), en la que se solicitaron sus opiniones y valoraciones sobre cuestiones sensibles relacionadas con "aspectos vinculados al bajo rendimiento de algunos estudiantes", "las situaciones de los estudiantes en la asignatura" y "el dictado de la asignatura". Los aspectos abarcados fueron los siguientes: Conocimientos traídos del secundario, dificultades de los contenidos, N° de horas destinado al estudio, el modo en que se estudia, rendimiento que se espera tener, N° de horas destinado al dictado, el modo en que se enseña, la relación docente / alumno, otros. Los resultados muestran convergencias y divergencias entre las percepciones de los estudiantes y las orientaciones actuales de la investigación educativa.Departamento de Ciencias Exactas y Naturale

Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials

[EN] We draw a fundamental compendium of the most valuable results of the theory of summing linear operators and detail those that are not shared by known multilinear and polynomial extensions of absolutely summing linear operators. The lack of such results in the theory of non-linear summing operators justifies the introduction of a class of polynomials and multilinear operators that satisfies at once all related non-linear results. Surprisingly enough, this class, defined by means of a summing inequality, happens to be the well known ideal of composition with a summing operator.D. Pellegrino acknowledges with thanks the support of CNPq Grant 401735/2013-3-PVE (Linha 2)-Brazil. P. Rueda acknowledges with thanks the support of the Ministerio de Economia y Competitividad (Spain) MTM2011-22417. E. A. Sanchez Perez acknowledges with thanks the support of the Ministerio de Economia y Competitividad (Spain) MTM2012-36740-C02-02.Pellegrino, D.; Rueda, P.; Sánchez Pérez, EA. (2016). Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 110(1):285-302. https://doi.org/10.1007/s13398-015-0224-8S2853021101Achour, D., Dahia, E., Rueda, P., Sánchez-Pérez, E.A.: Factorization of absolutely continuous polynomials. J. Math. Anal. Appl. 405(1), 259–270 (2013)Albiac, F., Kalton, N.: Topics in Banac Space Theory. Springer, Berlin (2005)Alencar, R., Matos, M.C.: Some classes of multilinear mappings between Banach spaces, Publicaciones del Departamento de Análisis Matemático 12, Universidad Complutense Madrid (1989)Bombal, F., Pérez-García, D., Villanueva, I.: Multilinear extensions of Grothendieck’s theorem. Q. J. Math. 55(4), 441–450 (2004)Botelho, G., Braunss, H.-A., Junek, H., Pellegrino, D.: Holomorphy types and ideals of multilinear mappings. Studia Math. 177, 43–65 (2006)Botelho, G., Pellegrino, D.: Scalar-valued dominated polynomials on Banach spaces. Proc. Am. Math. Soc. 134, 1743–1751 (2006)Botelho, G., Pellegrino, D.: Absolutely summing polynomials on Banach spaces with unconditional basis. J. Math. Anal. Appl. 321, 50–58 (2006)Botelho, G., Pellegrino, D.: Coincidence situations for absolutely summing non-linear mappings. Port. Math. (N.S.) 64(2), 175–191 (2007)Botelho, G., Pellegrino, D., Rueda, P.: Pietsch’s factorization theorem for dominated polynomials. J. Funct. Anal. 243(1), 257–269 (2007)Botelho, G., Pellegrino, D., Rueda, P.: On composition ideals of multilinear mappings and homogeneous polynomials. Publ. Res. Inst. Math. Sci. 43(4), 1139–1155 (2007)Botelho, G., Pellegrino, D., Rueda, P.: A unified Pietsch domination theorem. J. Math. Anal. Appl. 365, 269–276 (2010)Botelho, G., Pellegrino, D., Rueda, P.: Dominated polynomials on infinite dimensional spaces. Proc. Am. Math. Soc. 138(1), 209–216 (2010)Botelho, G., Pellegrino, D., Rueda, P.: Cotype and absolutely summing linear operators. Math. Z. 267(1–2), 1–7 (2011)Botelho, G., Pellegrino, D., Rueda, P.: On Pietsch measures for summing operators and dominated polynomials. Linear Multilinear Algebra 62(7), 860–874 (2014)Çalışkan, E., Pellegrino, D.M.: On the multilinear generalizations of the concept of absolutely summing operators. Rocky Mountain J. Math. 37, 1137–1154 (2007)Carando, D., Dimant, V.: On summability of bilinear operators. Math. Nachr. 259, 3–11 (2003)Carando, D., Dimant, V., Muro, S.: Coherent sequences of polynomial ideals on Banach spaces. Math. Nachr. 282, 1111–1133 (2009)Defant, A., Floret, K.: Tensor norms and operator ideals. North-Holland Mathematics Studies, 176. North-Holland Publishing Co., Amsterdam (1993)Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge University Press, Cambridge (1995)Dimant, V.: Strongly $$p$$ p -summing multilinear operators. J. Math. Anal. Appl. 278, 182–193 (2003)Dineen, S.: Complex analysis on infinite-dimensional spaces. Springer, London (1999)Fabian, M., Hájek, P., Montesinos-Santalucía, V., Pelant, J., Zizler, V.: Functional analysis and infinite-dimensional geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8. Springer, New York (2001)Floret, K.: Natural norms on symmetric tensor products of normed spaces. Note Mat. 17, 153–188 (1997)Geiss, H.: Ideale multilinearer Abbildungen. Diplomarbeit, Brandenburgische Landeshochschule (1985)Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques (French). Bol. Soc. Mat. São Paulo 8, 1–79 (1953)Jarchow, H., Palazuelos, C., Pérez-García, D., Villanueva, I.: Hahn-Banach extension of multilinear forms and summability. J. Math. Anal. Appl. 336, 1161–1177 (2007)Lindenstrauss, J., Pełczyński, A.: Absolutely summing operators in $${\cal L}_{p}$$ L p spaces and their applications. Studia Math. 29, 275–326 (1968)Matos, M.C.: Absolutely summing holomorphic mappings. Anais da Academia Brasileira de Ciências 68, 1–13 (1996)Matos, M.C.: Fully absolutely summing and Hilbert–Schmidt multilinear mappings. Collectanea Math. 54, 111–136 (2003)Matos, M.C.: Nonlinear absolutely summing mappings. Math. Nachr. 258, 71–89 (2003)Meléndez, Y., Tonge, A.: Polynomials and the Pietsch domination theorem. Proc. R. Irish Acad. Sect. A 99, 195–212 (1999)Montanaro, A.: Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53(12), 122–206 (2012)Mujica, J.: Complex analysis in Banach spaces. Dover Publications, Mineola (2010)Pellegrino, D.: Cotype and absolutely summing homogeneous polynomials in $${\cal L}_{p}$$ L p spaces. Studia Math. 157, 121–131 (2003)Pellegrino, D., Ribeiro, J.: On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility. Monatsh. Math. 173(3), 379–415 (2014)Pellegrino, D., Santos, J.: A general Pietsch domination theorem. J. Math. Anal. Appl. 375, 371–374 (2011)Pellegrino, D., Santos, J.: Absolutely summing multilinear operators: a panorama. Quaest. Math. 34(4), 447–478 (2011)Pellegrino, D., Santos, J.: On summability of nonlinear mappings: a new approach. Math. Z. 270(1–2), 189–196 (2012)Pellegrino, D., Santos, J., Seoane-Sepúlveda, J.B.: Some techniques on nonlinear analysis and applications. Adv. Math. 229, 1235–1265 (2012)Pérez-García, D.: David Comparing different classes of absolutely summing multilinear operators. Arch. Math. (Basel) 85(3), 258–267 (2005)Pietsch, A.: Absolut p-summierende Abbildungen in normierten Räumen. (German) Studia Math. 28, 333–353 (1966/1967)Pietsch, A.: Ideals of multilinear functionals (designs of a theory). Proceedings of the second international conference on operator algebras, ideals, and their applications in theoretical physics (Leipzig, 1983), 185–199, Teubner-Texte Math., 67, Teubner, Leipzig, (1984)Pisier, G.: Grothendieck’s theorem, past and present. Bull. Am. Math. Soc. (N.S.) 49(2), 237–323 (2012)Rueda, P., Sánchez-Pérez, E.A.: Factorization of $$p$$ p -dominated polynomials through $$L^{p}$$ L p -spaces. Michigan Math. J. 63(2), 345–353 (2014)Rueda, P., Sánchez-Pérez, E.A.: Factorization theorems for homogeneous maps on Banach function spaces and approximation of compact operators. Mediterr. J. Math. 12(1), 89–115 (2015)Ryan, R.A.: Applications of Topological Tensor Products to Infinite Dimensional Holomorphy, Ph.D. Thesis, Trinity College, Dublin, (1980

Unbounded violation of tripartite Bell inequalities

We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck's constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized GHZ states is always bounded so that, in contrast to many other contexts, GHZ states do in this case not lead to extremal quantum correlations. The results are based on tools from the theories of operator spaces and tensor norms which we exploit to prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras.Comment: Substantial changes in the presentation to make the paper more accessible for a non-specialized reade

Weighted Banach spaces of harmonic functions

“The final publication is available at Springer via http://dx.doi.org/10.1007/s13398-012-0109-z."We study Banach spaces of harmonic functions on open sets of or endowed with weighted supremum norms. We investigate the harmonic associated weight defined naturally as the analogue of the holomorphic associated weight introduced by Bierstedt, Bonet, and Taskinen and we compare them. We study composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions characterizing the continuity, the compactness and the essential norm of composition operators among these spaces in terms of associated weights.The research of the first author was partially supported by MEC and FEDER Project MTM2010-15200 and by GV project ACOMP/2012/090.Jorda Mora, E.; Zarco García, AM. (2014). Weighted Banach spaces of harmonic functions. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 108(2):405-418. https://doi.org/10.1007/s13398-012-0109-zS4054181082Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, Berlin (2001)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. J. 40(2), 271–297 (1993)Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127(2), 137–168 (1998)Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 54(1), 70–79 (1993)Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., Domański, P., Lindström, M.: Weakly compact composition operators on weighted vector-valued Banach spaces of analytic mappings. Ann. Acad. Sci. Fenn. Math. Ser. A I 26, 233–248 (2001)Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 64, 101–118 (1998)Bonet, J., Friz, M., Jordá, E.: Composition operators between weighted inductive limits of spaces of holomorphic functions. Publ. Math. Debr. Ser. A 67, 333–348 (2005)Boyd, C., Rueda, P.: The v-boundary of weighted spaces of holomorphic functions. Ann. Acad. Sci. Fenn. Math. 30, 337–352 (2005)Boyd, C., Rueda, P.: Complete weights and v-peak points of spaces of weighted holomorphic functions. Isr. J. Math. 155, 57–80 (2006)Boyd, C., Rueda, P.: Isometries of weighted spaces of harmonic functions. Potential Anal. 29(1), 37–48 (2008)Carando, D., Sevilla-Peris, P.: Spectra of weighted algebras of holomorphic functions. Math. Z. 263, 887–902 (2009)Contreras, M.D., Hernández-Díaz, G.: Weighted composition operators in weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 69(1), 41–60 (2000)García, D., Maestre, M., Rueda, P.: Weighted spaces of holomorphic functions on Banach spaces. Stud. Math. 138(1), 1–24 (2000)García, D., Maestre, M., Sevilla-Peris, P.: Composition operators between weighted spaces of holomorphic functions on Banach spaces. Ann. Acad. Sci. Fenn. Math. 29, 81–98 (2004)Gunning, R., Rossi, H.: Analytic Functions of Several Complex Variables. AMS Chelsea Publishing, Providence (2009)Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs (1962)Krantz, S.G.: Function Theory of Several Complex Variables. AMS, Providence (2001)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175(1), 19–45 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford University Press, Oxford (1997)Montes-Rodríguez, A.: Weight composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(2), 872–884 (2000)Ng, K.F.: On a theorem of Diximier. Math. Scand. 29, 279–280 (1972)Rudin, W.: Real and Complex Analysis. MacGraw-Hill, NY (1970)Rudin, W.: Functional analysis. In: International series in pure and applied mathematics, 2nd edn. McGraw-Hill, Inc., New York (1991)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of harmonic functions. J. Reine Angew. Math. 299(300), 256–279 (1978)Shields, A.L., Williams, D.L.: Bounded projections and the growth of harmonic conjugates in the unit disc. Mich. Math. J. 29, 3–25 (1982)Zheng, L.: The essential norms and spectra of composition operators on $$H^\infty$$ . Pac. J. Math. 203(2), 503–510 (2002

Procyanidin B3 Prevents Articular Cartilage Degeneration and Heterotopic Cartilage Formation in a Mouse Surgical Osteoarthritis Model

Osteoarthritis (OA) is a common disease in the elderly due to an imbalance in cartilage degradation and synthesis. Heterotopic ossification (HO) occurs when ectopic masses of endochondral bone form within the soft tissues around the joints and is triggered by inflammation of the soft tissues. Procyanidin B3 (B3) is a procyanidin dimer that is widely studied due to its high abundance in the human diet and antioxidant activity. Here, we evaluated the role of B3 isolated from grape seeds in the maintenance of chondrocytes in vitro and in vivo. We observed that B3 inhibited H2O2-induced apoptosis in primary chondrocytes, suppressed H2O2- or IL-1ß−induced nitric oxide synthase (iNOS) production, and prevented IL-1ß−induced suppression of chondrocyte differentiation marker gene expression in primary chondrocytes. Moreover, B3 treatment enhanced the early differentiation of ATDC5 cells. To examine whether B3 prevents cartilage destruction in vivo, OA was surgically induced in C57BL/6J mice followed by oral administration of B3 or vehicle control. Daily oral B3 administration protected articular cartilage from OA and prevented chondrocyte apoptosis in surgically-induced OA joints. Furthermore, B3 administration prevented heterotopic cartilage formation near the surgical region. iNOS protein expression was enhanced in the synovial tissues and the pseudocapsule around the surgical region in OA mice fed a control diet, but was reduced in mice that received B3. Together, these data indicated that in the OA model, B3 prevented OA progression and heterotopic cartilage formation, at least in a part through the suppression of iNOS. These results support the potential therapeutic benefits of B3 for treatment of human OA and heterotopic ossification

Atomic decompositions for tensor products and polynomial spaces

We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product ⊗s, μ n X, for any symmetric tensor norm μ. In addition, the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained. Finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on X. © 2008 Elsevier Inc. All rights reserved.Fil:Carando, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.Fil:Lassalle, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina