Factorizing Kernel Operators

Consider an operator T : X--->Y between Banach function spaces having adequate order continuity and Fatou properties. Assume that T can be factorized through a Banach space as T =SR, where R and the adjoint of S are p-th power and q-th power factorable, respectively. Then a canonical factorization scheme can be given for T. We show that it provides a tool for analyzing T that becomes specially useful for the case of kernel operators. In particular, we show that this square factorization scheme for T is equivalent to some inequalities for the bilinear form defined by T. Kernel operators are studied from this point of view.Support of the Ministerio de Economia y Competitividad under project # MTM2012-36740-C02-02 (Spain) is gratefully acknowledged.Galdames, O.; Sánchez Pérez, EA. (2013). Factorizing Kernel Operators. Integral Equations and Operator Theory. 75(1):13-29. https://doi.org/10.1007/s00020-012-2019-zS1329751Bennett C., Sharpley R.: Interpolation of Operators. Academic Press, Boston (1988)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)Defant A.: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant A., Sánchez Pérez E.A.: Maurey-Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1977)Fernández A., Mayoral F., Naranjo F., Sáez C., Sánchez-Pérez E.A.: Spaces of integrable functions with respect to a vector measure and factorizations through L p and Hilbert spaces. J. Math. Anal. Appl. 330, 1249–1263 (2007)Galdames O., Sánchez Pérez E.A.: Optimal range theorems for operators with p-th power factorable adjoints. Banach J. Math. Anal. 6(1), 63–71 (2012)Lindenstrauss J., Tzafriri L.: Classical Banach Spaces II. Springer, Berlin (1979)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)Zaanen, A.C.: Integration, 2nd revised edn. North Holland, Amsterdam; Interscience, New York (1967