Extrapolation theorems for \((p,q)\)-factorable operators. (English) Zbl 1496.47036

Galdames-Bravo, Orlando

Banach J. Math. Anal. 12, No. 2, 497-514 (2018).

Summary: The operator ideal of \((p,q)\)-factorable operators can be characterized as the class of operators that factors through the embedding \(L^{q'}(\mu)\hookrightarrow L^{p}(\mu)\) for a finite measure \(\mu\), where \(p,q\in[1,\infty)\) are such that \(1/p+1/q\geq1\). We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through \(r\)th and \(s\)th power factorable operators, for suitable \(r,s\in[1,\infty)\). Thus, they also factor through a positive map \(L^{s}(m_{1})^{\ast}\to L^{r}(m_{2})\), where \(m_{1}\) and \(m_{2}\) are vector measures. We use the properties of the spaces of \(u\)-integrable functions with respect to a vector measure and the \(u\)th power factorable operators to obtain a characterization of \((p,q)\)-factorable operators and conditions under which a \((p,q)\)-factorable operator is \(r\)-summing for \(r\in[1,p]\).

MSC:

47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46G10	Vector-valued measures and integration

Keywords:

\((p,q)\)-factorable operator; pth power factorable operator; vector measure

References:

[1]	O. Blasco, J. M. Calabuig, and E. A. Sánchez-Pérez, \(p\)-variations of vector measures with respect to vector measures and integral representation of operators, Banach J. Math. Anal. 9 (2015), no. 1, 273-285. · Zbl 1337.46034
[2]	J. M. Calabuig, O. Delgado, and E. A. Sánchez-Pérez, Factorizing operators on Banach function spaces through spaces of multiplication operators, J. Math. Anal. Appl. 364 (2010), no. 1, 88-103. · Zbl 1191.47023 · doi:10.1016/j.jmaa.2009.10.034
[3]	G. P. Curbera, When \(L^1\) of a vector measure is an AL-space, Pacific J. Math. 162 (1994), no. 2, 287-303. · Zbl 0791.46021
[4]	A. Defant, Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces, Positivity 5 (2001), no. 2, 153-175. · Zbl 0994.47036
[5]	A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, Amsterdam, 1993. · Zbl 0774.46018
[6]	J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, Cambridge, 1995. · Zbl 0855.47016
[7]	J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys Monogr. 15, Amer. Math. Soc., Providence, 1977. · Zbl 0369.46039
[8]	O. Galdames-Bravo and E. A. Sánchez-Pérez, Optimal range theorems for operators with \(p\) th power factorable adjoints, Banach J. Math. Anal. 6 (2012), no. 1, 61-73. · Zbl 1276.47003
[9]	O. Galdames-Bravo and E. A. Sánchez-Pérez, Factorizing kernel operators, Integral Equations Operator Theory 75 (2013), no. 1, 13-29. · Zbl 1342.46031
[10]	S. Kwapień, “On operators factorizable through \(L_p\) space” in Actes du Colloque d’Analyse Fonctionnelle (Bordeaux, 1971), Bull. Soc. Math. France 100, Soc. Math. France, Paris, 1972, 215-225. · Zbl 0246.47040
[11]	J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II: Function Spaces, Ergeb. Math. Grenzgeb. (3) 97, Springer, Berlin, 1979. · Zbl 0403.46022
[12]	B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces \(L^p\), Astérisque 11, Soc. Math. France, Paris, 1974. · Zbl 0278.46028
[13]	S. Okada, W. J. Ricker, and L. Rodríguez-Piazza, Compactness of the integration operator associated with a vector measure, Studia Math. 150 (2002), no. 2, 133-149. · Zbl 0998.28009
[14]	S. Okada, W. J. Ricker, and L. Rodríguez-Piazza, Operator ideal properties of vector measures with finite variation, Studia Math. 205 (2011), no. 3, 215-249. · Zbl 1236.28009
[15]	S. Okada, W. J. Ricker, and E. A. Sánchez-Pérez, Optimal Domain and Integral Extension of Operators: Acting in Function Spaces, Oper. Theory Adv. Appl. 180, Birkhäuser, Basel, 2008. · Zbl 1145.47027
[16]	A. Pietsch, Operator Ideals, North-Holland Math. Libr. 20, North-Holland, Amsterdam, 1979.