Optimal Extensions for pth Power Factorable Operators

[EN] Let be a function space related to a measure space with and let be a Banach space-valued operator. It is known that if T is pth power factorable then the largest function space to which T can be extended preserving pth power factorability is given by the space L (p) (m (T) ) of p-integrable functions with respect to m (T) , where is the vector measure associated to T via . In this paper, we extend this result by removing the restriction . In this general case, by considering m (T) defined on a certain -ring, we show that the optimal domain for T is the space . We apply the obtained results to the particular case when T is a map between sequence spaces defined by an infinite matrix.O. Delgado gratefully acknowledges the support of the Ministerio de Economia y Competitividad (Project #MTM2012-36732-C03-03) and the Junta de Andalucia (Projects FQM-262 and FQM-7276), Spain. E.A. Sanchez Perez acknowledges with thanks the support of the Ministerio de Economia y Competitividad (Project #MTM2012-36740-C02-02), Spain.Delgado Garrido, O.; Sánchez Pérez, EA. (2016). Optimal Extensions for pth Power Factorable Operators. Mediterranean Journal of Mathematics. 13(6):4281-4303. https://doi.org/10.1007/s00009-016-0745-142814303136Bennett, C., Sharpley, R.: Interpolation of operators. Academic Press, Boston (1988)Brooks 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., Juan M.A., Sánchez Pérez E.A.: On the Banach lattice structure of L 1 w of a vector measure on a $${\delta}$$ δ -ring. Collect. Math 65, 67–85 (2014)Calabuig J.M., Delgado J.M., 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 $${\delta}$$ δ -ring. Operators Matrices 6, 241–262 (2012)del Campo R., Fernández A., Galdames O., Mayoral F., Naranjo F.: Complex interpolation of operators and optimal domains. Integr. Equ. Oper. Theory 80, 229–238 (2014)Delgado O.: L 1-spaces of vector measures defined on $${\delta}$$ δ -rings. Arch. Math. 84, 432–443 (2005)Delgado O.: Optimal domains for kernel operators on $${[0,\infty)\times[0,\infty)}$$ [ 0 , ∞ ) × [ 0 , ∞ ) . Studia Math. 174, 131–145 (2006)Diestel, J., Uhl Jr, J.J.: Vector measures. Math. Surveys, vol. 15. American Mathematical Society, Providence (1977)Galdames Bravo O.: On the norm with respect to vector measures of the solution of an infinite system of ordinary differential equations. Mediterr. J. Math. 12, 939–956 (2015)Galdames Bravo O.: Generalized Köthe p-dual spaces. Bull. Belg. Math. Soc. Simon Stevin 21, 275–289 (2014)Galdames Bravo O., Sánchez Pérez E.A.: Optimal range theorems for operators with p-th power factorable adjoints. Banach J. Math. Anal. 6, 61–73 (2012)Galdames Bravo O., Sánchez Pérez E.A.: Factorizing kernel operators. Integr. Equ. Oper. Theory 75, 13–29 (2013)Kalton, N.J., Peck, N.T., Roberts, J.W.: An F-space Sampler. London Math. Soc. Lecture Notes, vol. 89. Cambridge University Press, Cambridge (1985)Lewis D.R: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, vol.II. Springer, Berlin (1979)Masani P.R., Niemi H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on $${\delta}$$ δ -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)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