Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces

We consider the problem of extending or factorizing a bounded bilinear map defined on a couple of order continuous Banach function spaces to its optimal domain, i.e. the biggest couple of Banach function spaces to which the bilinear map can be extended. As in the case of linear operators, we use vector measure techniques to find this space, and we show that this procedure cannot be always successfully used for bilinear maps. We also present some applications to find optimal factorizations of linear operators between Banach function spaces.J. M. Calabuig was supported by Ministerio de Economia y Competitividad (Spain) (project MTM2011-23164) and by "Jose Castillejo 2009" (MEC). E. A. Sanchez-Perez was supported by MEC and FEDER (project MTM2009-14483-C02-02). J. M. Calabuig and E. A. Sanchez-Perez were also supported by Ayuda para Estancias de PDI de la UPV en Centros de Investigacion de Prestigio (PAID-00-11).Calabuig Rodriguez, JM.; Fernandez Unzueta, M.; Galaz Fontes, F.; Sánchez Pérez, EA. (2014). Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas (RACSAM). 108(2):353-367. https://doi.org/10.1007/s13398-012-0101-7S3533671082Calabuig, J.M., Galaz-Fontes, F., Jiménez Fernández, E., Sánchez Pérez, E.A.: Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series. Math. Z. 257, 381–402 (2007)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Curbera, G.P.: Operators into $$L^1$$ of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47–63 (2002)Curbera, G.P. , Ricker, W.J.: Optimal domains for the kernel operator associated with Sobolev’s inequality. Studia Math. 158(2), 131–152 (2003) [see also Corrigenda in the same journal, 170 (2005) 217–218)]Delgado, O.: Banach function subspaces of $$L^1$$ of a vector measure and related Orlicz spaces. Indag. Math. (N. S.) 15, 485–495 (2004)Delgado, O.: Optimal domains for kernel operators on $$[0,\infty )\times [0,\infty )$$ . Studia Math. 174, 131–145 (2006)Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244, 119–133 (2007)Diestel, J., Uhl, J.J.: Vector measures. In: Math. Surveys, vol. 15. Amer. Math. Soc., Providence (1977)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Galdames, O., Sánchez Pérez, E.A.: Optimal range theorems for operators with $$p$$ -th power factorable adjoints. Banach J. Math. Anal. 6(1), 61–73 (2012)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. In: Oper. Theory Adv. Math. Appl., vol. 180. Birkäuser, Basel (2008

On the Banach lattice structure of L-w(1) of a vector measure on a delta-ring

We study some Banach lattice properties of the space L-w(1)(v) of weakly integrable functions with respect to a vector measure v defined on a delta-ring. Namely, we analyze order continuity, order density and Fatou type properties. We will see that the behavior of L-w(1)(v) differs from the case in which is defined on a sigma-algebra whenever does not satisfy certain local sigma-finiteness property.J. M. Calabuig and M. A. Juan were supported by the Ministerio de Economia y Competitividad (project MTM2008-04594). O. Delgado was supported by the Ministerio de Economia y Competitividad (project MTM2009-12740-C03-02). E. A. Sanchez Perez was supported by the Ministerio de Economia y Competitividad (project MTM2009-14483-C02-02).Calabuig Rodriguez, JM.; Delgado Garrido, O.; Juan Blanco, MA.; Sánchez Pérez, EA. (2014). On the Banach lattice structure of L-w(1) of a vector measure on a delta-ring. Collectanea Mathematica. 65(1):67-85. doi:10.1007/s13348-013-0081-8S6785651Brooks, J.K., Dinculeanu, N.: Strong additivity, absolute continuity and compactness in spaces of measures. J. Math. Anal. Appl. 45, 156–175 (1974)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Calabuig, J.M., Juan, M.A., Sánchez Pérez, E.A.: Spaces of $$p$$ -integrable functions with respect to a vector measure defined on a $$\delta$$ -ring. Oper. Matrices 6, 241–262 (2012)Curbera, G.P.: El espacio de funciones integrables respecto de una medida vectorial. Ph. D. thesis, University of Sevilla, Sevilla (1992)Curbera, G.P.: Operators into $$L^1$$ of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. (N.S.) 17, 187–204 (2006)G. P. Curbera and W. J. Ricker, Vector measures, integration and applications. In: Positivity (in Trends Math.), Birkhäuser, Basel, pp. 127–160 (2007)Curbera, G.P., Ricker, W.J.: The Fatou property in $$p$$ -convex Banach lattices. J. Math. Anal. Appl. 328, 287–294 (2007)Delgado, O.: $$L^1$$ -spaces of vector measures defined on $$\delta$$ -rings. Arch. Math. 84, 432–443 (2005)Delgado, O.: Optimal domains for kernel operators on $$[0,\infty )\times [0,\infty )$$ . Studia Math. 174, 131–145 (2006)Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244, 119–133 (2007)Delgado, O., Juan, M.A.: Representation of Banach lattices as $$L_w^1$$ spaces of a vector measure defined on a $$\delta$$ -ring. Bull. Belg. Math. Soc. Simon Stevin 19(2), 239–256 (2012)Diestel, J., Uhl, J.J.: Vector measures (Am. Math. Soc. surveys 15). American Mathematical Society, Providence (1997)Dinculeanu, N.: Vector measures, Hochschulbcher fr Mathematik, vol. 64. VEB Deutscher Verlag der Wissenschaften, Berlin (1966)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)Fremlin, D.H.: Measure theory, broad foundations, vol. 2. Torres Fremlin, Colchester (2001)Jiménez Fernández, E., Juan, M.A., Sánchez Pérez, E.A.: A Komlós theorem for abstract Banach lattices of measurable functions. J. Math. Anal. Appl. 383, 130–136 (2011)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Luxemburg, W.A.J., Zaanen, A.C.: Riesz spaces I. North-Holland, Amsterdam (1971)Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on $$\delta$$ -rings. Adv. Math. 73, 204–241 (1989)Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 75, 121–167 (1989)Thomas, E.G.F.: Vector integration (unpublished) (2013)Turpin, Ph.: Intégration par rapport à une mesure à valeurs dans un espace vectoriel topologique non supposé localement convexe, Intègration vectorielle et multivoque, (Colloq., University Caen, Caen, 1975), experiment no. 8, Dèp. Math., UER Sci., University Caen, Caen (1975)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces (Oper. Theory Adv. Appl.), vol. 180. Birkhäuser, Basel (2008)Zaanen, A.C.: Riesz spaces II. North-Holland, Amsterdam (1983

Mean ergodicity and spectrum of the Cesàro operator on weighted c0 spaces

[EN] A detailed investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator C acting on the weighted Banach sequence space c0(w) for a bounded, strictly positive weight w. New features arise in the weighted setting (e.g. existence of eigenvalues, compactness, mean ergodicity) which are not present in the classical setting of c0.The research of the first two authors was partially supported by the Projects MTM2013-43540-P, GVA Prometeo II/2013/013 and ACOMP/2015/186 (Spain).Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2016). Mean ergodicity and spectrum of the Cesàro operator on weighted c0 spaces. Positivity. 20:761-803. https://doi.org/10.1007/s11117-015-0385-xS76180320Akhmedov, A.M., Başar, F.: On the fine spectrum of the Cesàro operator in $$c_0$$ c 0 . Math. J. Ibaraki Univ. 36, 25–32 (2004)Akhmedov, A.M., Başar, F.: The fine spectrum of the Cesàro operator $$C_1$$ C 1 over the sequence space $$bv_p, (1 \le p < \infty )$$ b v p , ( 1 ≤ p < ∞ ) . Math. J. Okayama Univ. 50, 135–147 (2008)Albanese, A.A., Bonet, J., Ricker, W.J.: Convergence of arithmetic means of operators in Fréchet spaces. J. Math. Anal. Appl. 401, 160–173 (2013)Albanese, A.A., Bonet, J., Ricker, W.J.: Spectrum and compactness of the Cesàro operator on weighted $$\ell _p$$ ℓ p spaces. J. Aust. Math. Soc. 99, 287–314 (2015)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in the Fréchet spaces $$\ell ^{p+}$$ ℓ p + and $$L ^{p-}$$ L p - . Glasg. Math. J (to appear)Ansari, S.I., Bourdon, P.S.: Some properties of cyclic operators. Acta Sci. Math. Szeged 63, 195–207 (1997)Brown, A., Halmos, P.R., Shields, A.L.: Cesàro operators. Acta Sci. Math. Szeged 26, 125–137 (1965)Curbera, G.P., Ricker, W.J.: Spectrum of the Cesàro operator in $$\ell ^p$$ ℓ p . Arch. Math. 100, 267–271 (2013)Curbera, G.P., Ricker, W.J.: Solid extensions of the Cesàro operator on $$\ell ^p$$ ℓ p and $$c_0$$ c 0 . Integr. Equ. Oper. Theory 80, 61–77 (2014)Curbera, G.P., Ricker, W.J.: The Cesàro operator and unconditional Taylor series in Hardy spaces. Integr. Equ. Oper. Theory 83, 179–195 (2015)Diestel, J.: Sequences and Series in Banach Spaces. Springer, New York (1984)Dowson, H.R.: Spectral Theory of Linear Operators. Academic Press, London (1978)Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory, 2nd Printing. Wiley Interscience Publ, New York (1964)Emilion, R.: Mean-bounded operators and mean ergodic theorems. J. Funct. Anal. 61, 1–14 (1985)Goldberg, S.: Unbounded Linear Operators: Theory and Applications. Dover Publ, New York (1985)Hille, E.: Remarks on ergodic theorems. Trans. Am. Math. Soc. 57, 246–269 (1945)Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)Krengel, U.: Ergodic Theorems. de Gruyter, Berlin (1985)Leibowitz, G.: Spectra of discrete Cesàro operators. Tamkang J. Math. 3, 123–132 (1972)Lin, M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 337–340 (1974)Megginson, R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)Mureşan, M.: A Concrete Approach to Classical Analysis. Springer, Berlin (2008)Okutoyi, J.I.: On the spectrum of $$C_1$$ C 1 as an operator on $$bv_0$$ b v 0 . J. Aust. Math. Soc. Ser. A 48, 79–86 (1990)Radjavi, H., Tam, P.-W., Tan, K.-K.: Mean ergodicity for compact operators. Studia Math. 158, 207–217 (2003)Reade, J.B.: On the spectrum of the Cesàro operator. Bull. Lond. Math. Soc. 17, 263–267 (1985)Rhoades, B.E., Yildirim, M.: The spectra and fine spectra of factorable matrices on $$c_0$$ c 0 . Math. Commun. 16, 265–270 (2011)Taylor, A.E.: Introduction to Functional Analysis. Wiley, New York (1958

Maurey-Rosenthal domination for abstract Banach lattices

We extend the Maurey-Rosenthal theorem on integral domination and factorization of p-concave operators from a p-convex Banach function space through Lp-spaces for the case of operators on abstract p-convex Banach lattices satisfying some essential lattice requirements - mainly order density of its order continuous part - that are shown to be necessary. We prove that these geometric properties can be characterized by means of an integral inequality giving a domination of the pointwise evaluation of the operator for a suitable weight also in the case of abstract Banach lattices. We obtain in this way what in a sense can be considered the most general factorization theorem of operators through Lp-spaces. In order to do this, we prove a new representation theorem for abstract p-convex Banach lattices with the Fatou property as spaces of p-integrable functions with respect to a vector measure.The authors are supported by grants MTM2011-23164 and MTM2012-36740-C02-02 of the Ministerio de Economia y Competitividad (Spain).Juan Blanco, MA.; Sánchez Pérez, EA. (2013). Maurey-Rosenthal domination for abstract Banach lattices. Journal of Inequalities and Applications. (213). https://doi.org/10.1186/1029-242X-2013-213S213Defant A: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 2001, 5: 153–175. 10.1023/A:1011466509838Defant A, Sánchez Pérez EA: Maurey-Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 2004, 297: 771–790. 10.1016/j.jmaa.2004.04.047Defant A, Sánchez Pérez EA: Domination of operators on function spaces. Math. Proc. Camb. Philos. Soc. 2009, 146: 57–66. 10.1017/S0305004108001734Fernández A, Mayoral F, Naranjo F, Sáez C, Sánchez-Pérez EA: Vector measure Maurey-Rosenthal type factorizations and l -sums of L 1 -spaces. J. Funct. Anal. 2005, 220: 460–485. 10.1016/j.jfa.2004.06.010Palazuelos C, Sánchez Pérez EA, Tradacete P: Maurey-Rosenthal factorization for p -summing operators and Dodds-Fremlin domination. J. Oper. Theory 2012, 68(1):205–222.Luxemburg WAJ, Zaanen AC: Riesz Spaces I. North-Holland, Amsterdam; 1971.Zaanen AC: Riesz Spaces II. North-Holland, Amsterdam; 1983.Lindenstrauss J, Tzafriri L: Classical Banach Spaces II. Springer, Berlin; 1979.Aliprantis CD, Burkinshaw O: Positive Operators. Academic Press, New York; 1985.Curbera GP, Ricker WJ: Vector measures, integration and applications. Trends Math. In Positivity. Birkhäuser, Basel; 2007:127–160.Okada S, Ricker WJ, Sánchez Pérez EA: Optimal domains and integral extensions of operators acting in function spaces. 180. In Operator Theory Advances and Applications. Birkhäuser, Basel; 2008.Delgado O: L 1 -spaces of vector measures defined on δ -rings. Arch. Math. 2005, 84: 432–443. 10.1007/s00013-005-1128-1Calabuig, JM, Delgado, O, Juan, MA, Sánchez Pérez, EA: On the Banach lattice structure of L w 1 of a vector measure on a δ-ring. Collect. Math. doi:10.1007/s13348–013–0081–8Calabuig JM, Delgado O, Sánchez Pérez EA: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 2010, 364: 88–103. 10.1016/j.jmaa.2009.10.034Delgado O:Optimal domains for kernel operators on [ 0 , ∞ ) × [ 0 , ∞ ) .Stud. Math. 2006, 174: 131–145. 10.4064/sm174-2-2Delgado O, Soria J: Optimal domain for the Hardy operator. J. Funct. Anal. 2007, 244: 119–133. 10.1016/j.jfa.2006.12.011Jiménez Fernández E, Juan MA, Sánchez Pérez EA: A Komlós theorem for abstract Banach lattices of measurable functions. J. Math. Anal. Appl. 2011, 383: 130–136. 10.1016/j.jmaa.2011.05.010Curbera, GP: El espacio de funciones integrables respecto de una medida vectorial. PhD thesis, Univ. of Sevilla (1992)Sánchez Pérez EA: Compactness arguments for spaces of p -integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Ill. J. Math. 2001, 45(3):907–923.Fernández A, Mayoral F, Naranjo F, Sáez C, Sánchez-Pérez EA: Spaces of p -integrable functions with respect to a vector measure. Positivity 2006, 10: 1–16. 10.1007/s11117-005-0016-zCalabuig JM, Juan MA, Sánchez Pérez EA: Spaces of p -integrable functions with respect to a vector measure defined on a δ -ring. Oper. Matrices 2012, 6: 241–262.Lewis DR: On integrability and summability in vector spaces. Ill. J. Math. 1972, 16: 294–307.Masani PR, Niemi H: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on δ -rings. Adv. Math. 1989, 73: 204–241. 10.1016/0001-8708(89)90069-8Masani PR, Niemi H: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 1989, 75: 121–167. 10.1016/0001-8708(89)90035-2Brooks JK, Dinculeanu N: Strong additivity, absolute continuity and compactness in spaces of measures. J. Math. Anal. Appl. 1974, 45: 156–175. 10.1016/0022-247X(74)90130-9Curbera GP:Operators into L 1 of a vector measure and applications to Banach lattices.Math. Ann. 1992, 293: 317–330. 10.1007/BF01444717Delgado O, Juan MA: Representation of Banach lattices as L w 1 spaces of a vector measure defined on a δ -ring. Bull. Belg. Math. Soc. Simon Stevin 2012, 19: 239–256.Curbera GP, Ricker WJ: Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. 2006, 17: 187–204. 10.1016/S0019-3577(06)80015-7Curbera GP, Ricker WJ: The Fatou property in p -convex Banach lattices. J. Math. Anal. Appl. 2007, 328: 287–294. 10.1016/j.jmaa.2006.04.086Aliprantis CD, Border KC: Infinite Dimensional Analysis. Springer, Berlin; 1999.Delgado, O: Optimal extension for positive order continuous operators on Banach function spaces. Glasg. Math. J. (to appear

Kothe dual of Banach lattices generated by vector measures

We study the Kothe dual spaces of Banach function lattices generated by abstract methods having roots in the theory of interpolation spaces. We apply these results to Banach spaces of integrable functions with respect to Banach space valued countably additive vector measures. As an application we derive a description of the Banach dual of a large class of these spaces, including Orlicz spaces of integrable functions with respect to vector measuresThe first author was supported by the Foundation for Polish Science (FNP). The second author was supported by the Ministerio de Economia y Competitividad (Spain) under Grant #MTM2012-36740-C02-02.Mastylo, M.; Sánchez Pérez, EA. (2014). Kothe dual of Banach lattices generated by vector measures. Monatshefte fur Mathematik. 173(4):541-557. https://doi.org/10.1007/s00605-013-0560-8S5415571734Aronszajn, N., Gagliardo, E.: Interpolation spaces and interpolation methods. Ann. Mat. Pura. Appl. 68, 51–118 (1965)Bartle, R.G., Dunford, N., Schwartz, J.: Weak compactness and vector measures. Canad. J. Math. 7, 289–305 (1955)Brudnyi, Yu.A., Krugljak, N.Ya.: Interpolation functors and interpolation spaces $$I$$ I . North-Holland, Amsterdam (1991)Curbera, G.P.: Operators into $$L^1$$ L 1 of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: The Fatou property in $$p$$ p -convex Banach lattices. J. Math. Anal. Appl. 328, 287–294 (2007)Delgado, O.: Banach function subspaces of $$L^1$$ L 1 of a vector measure and related Orlicz spaces. Indag. Math. 15(4), 485–495 (2004)Diestel, J., Jr., Uhl, J.J.: Vector measures, Amer. Math. Soc. Surveys 15, Providence, R.I. (1977)Fernández, A., Mayoral, F., Naranjo, F., Sánchez-Pérez, E.A.: Spaces of $$p$$ 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, 1439–1444 (2008)Ferrando, I., Sánchez Pérez, E.A.: Tensor product representation of the (pre)dual of the $$L_p$$ L p -space of a vector measure. J. Aust. Math. Soc. 87, 211–225 (2009)Galaz-Fontes, F.: The dual space of $$L^p$$ L p of a vector measure. Positivity 14(4), 715–729 (2010)Kamińska, A.: Indices, convexity and concavity in Musielak-Orlicz spaces, dedicated to Julian Musielak. Funct. Approx. Comment. Math. 26, 67–84 (1998)Kantorovich, L.V., Akilov, G.P.: Functional analysis, 2nd edn. Pergamon Press, New York (1982)Krein, S.G., Petunin, Yu.I., Semenov, E.M.: Interpolation of linear operators. In: Translations of mathematical monographs, 54. American Mathematical Society, Providence, R.I., (1982)Lewis, D.R.: Integration with respect to vector measures. Pacific. J. Math. 33, 157–165 (1970)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 583–599 (1973)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices, (Russian). Sibirsk. Mat. Z. 10, 419–430 (1969)Musielak, J.: Orlicz spaces and modular spaces. In: Lecture Notes in Math. 1034, Springer-Verlag, Berlin (1983)Okada, S.: The dual space of $$L^1(\mu )$$ L 1 ( μ ) of a vector measure $$\mu$$ μ . J. Math. Anal. Appl. 177, 583–599 (1993)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces, operator theory. Adv. Appl., vol. 180, Birkhäuser, Basel (2008)Rao, M.M., Zen, Z.D.: Applications of Orlicz spaces. Marcel Dekker, Inc., New York (2002)Rivera, M.J.: Orlicz spaces of integrable functions with respect to vector-valued measures. Rocky Mt. J. Math. 38(2), 619–637 (2008)Sánchez Pérez, E.A.: Compactness arguments for spaces of $$p$$ p -integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Ill. J. Math. 45(3), 907–923 (2001)Sánchez Pérez, E.A.: Vector measure duality and tensor product representation of $$L_p$$ L p spaces of vector measures. Proc. Amer. Math. Soc. 132, 3319–3326 (2004)Zaanen, A.C.: Integration. North Holland, Amsterdam (1967

Completability and optimal factorization norms in tensor products of Banach function spaces

[EN] Given s-finite measure spaces ( 1, 1, mu 1) and ( 2, 2, mu 2), we consider Banach spaces X1(mu 1) and X2(mu 2), consisting of L0(mu 1) and L0(mu 2) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product X1(mu 1). p X2(mu 2) is continuously included in the metric space of measurable functions L0(mu 1. mu 2). In particular, we prove that the elements of the completion of the projective tensor product of L p-spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally showthat given a bounded linear operator T : X1(mu 1). p X2(mu 2). E (where E is a Banach space), a norm can be found for T to be bounded, which is ` minimal' with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces.J. M. Calabuig and M. Fernandez-Unzueta were supported by Ministerio de Economia, Industria y Competitividad (Spain) under project MTM2014-53009-P. M. Fernandez-Unzueta was also suported by CONACyT 284110. F. Galaz-Fontes was supported by Ministerio de Ciencia e Innovacion (Spain) and FEDER under project MTM2009-14483-C02-01. E. A. Sanchez Perez was supported by Ministerio de Economia, Industria y Competitividad (Spain) and FEDER under project MTM2016-77054-C2-1-P.Calabuig, JM.; Fernández-Unzueta, M.; Galaz-Fontes, F.; Sánchez Pérez, EA. (2019). Completability and optimal factorization norms in tensor products of Banach function spaces. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):3513-3530. https://doi.org/10.1007/s13398-019-00711-7S351335301134Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics, Vol 50, AMS (2002)Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)Bu, Q., Buskes, G., Kusraev, A.G.: Bilinear maps on products of vector lattices: a survey. In: Boulabiar, K., Buskes, G., Triki, A. (eds.) Positivity-Trends in Mathematics. Birkhäser Verlag AG, Basel, pp. 97–26 (2007)Buskes, G., Van Rooij, A.: Bounded variation and tensor products of Banach lattices. Positivity 7, 47–59 (2003)Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F., Sánchez-Pérez, E.A.: Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. RACSAM 108(2), 353–367 (2014)Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F., Sánchez-Pérez, E.A.: Equivalent norms in a Banach function space and the subsequence property. J. Korean Math. Soc. https://doi.org/10.4134/JKMS.j180682Curbera, G.P., Ricker, W.J.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47–63 (2002)Curbera, G.P., Ricker, W.J.: Vector measures, integration and applications. In: Positivity. Birkhäuser Basel, pp. 127–160 (2007)Gil de Lamadrid, J.: Uniform cross norms and tensor products. J. Duke Math. 32, 797–803 (1965)Dunford, N., Schwartz, J.: Linear Operators, Part I: General Theory. Interscience Publishers Inc., New York (1958)Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94(3), 777–798 (1972)Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211(2), 87–106 (1974)Yew, K.L.: Completely $$p$$-summing maps on the operator Hilbert space OH. J. Funct. Anal. 255, 1362–1402 (2008)Kwapien, S., Pelczynski, A.: The main triangle projection in matrix spaces and its applications. Stud. Math. 34(1), 43–68 (1970)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland Publishing Company, Amsterdam (1971)Milman, M.: Some new function spaces and their tensor products. Depto. de Matemática, Facultad de Ciencias, U. de los Andes, Mérida, Venezuela (1978)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. Oper. Theory Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Schep, A.R.: Factorization of positive multilinear maps. Illinois J. Math. 579–591 (1984)Zaanen, A.C.: Integration. North-Holland Publishing Company, Amsterdam-New York (1967)Zaanen, A.C.: Riesz Spaces II. North-Holland Publishing Company, Amsterdam (1983

The ℓs\ell^s-boundedness of a family of integral operators on UMD Banach function spaces

We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia.Comment: 13 pages. Generalization of arXiv:1410.665

Operators on the Fréchet sequence space ces(p+), 1≤p<∞1 \leq p < \infty

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