There is no strictly singular centralizer on 𝐿_{𝑝}

Cabello Sánchez, Félix

doi:10.1090/S0002-9939-2013-11851-4

There is no strictly singular centralizer on $L_p$
HTML articles powered by AMS MathViewer

by Félix Cabello Sánchez PDF

Proc. Amer. Math. Soc. 142 (2014), 949-955 Request permission

Abstract:

We prove that if $\Phi$ is a centralizer on $L_p$, where $0<p<\infty$, then there is a copy of $\ell _2$ inside $L_p$ where $\Phi$ is bounded. If $\Phi$ is symmetric, then it is also bounded on a copy of $\ell _q$, provided $0<p<q<2$. This shows that for a wide class of exact sequences $0\to L_p\to Z\to L_p\to 0$ the quotient map is not strictly singular, which generalizes a recent result of Jesús Suárez.

References

Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
F. Cabello Sánchez, Nonlinear centralizers in homology, to appear in Math. Ann. DOI 10.1007/S00208-013-0942-1.
Félix Cabello Sánchez, Jesús M. F. Castillo, and Jesús Suárez, On strictly singular nonlinear centralizers, Nonlinear Anal. 75 (2012), no. 7, 3313–3321. MR 2891170, DOI 10.1016/j.na.2011.12.022
W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298. MR 527010, DOI 10.1090/memo/0217
N. J. Kalton, Applications of an inequality for $H_1$-functions, Texas Functional Analysis Seminar 1985–1986 (Austin, TX, 1985–1986) Longhorn Notes, Univ. Texas, Austin, TX, 1986, pp. 41–51. MR 1017041
Nigel J. Kalton, Nonlinear commutators in interpolation theory, Mem. Amer. Math. Soc. 73 (1988), no. 385, iv+85. MR 938889, DOI 10.1090/memo/0385
N. J. Kalton, Differentials of complex interpolation processes for Köthe function spaces, Trans. Amer. Math. Soc. 333 (1992), no. 2, 479–529. MR 1081938, DOI 10.1090/S0002-9947-1992-1081938-1
Nigel Kalton and Stephen Montgomery-Smith, Interpolation of Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1131–1175. MR 1999193, DOI 10.1016/S1874-5849(03)80033-5
N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1–30. MR 542869, DOI 10.1090/S0002-9947-1979-0542869-X
Richard Rochberg and Guido Weiss, Derivatives of analytic families of Banach spaces, Ann. of Math. (2) 118 (1983), no. 2, 315–347. MR 717826, DOI 10.2307/2007031
John V. Ryff, Orbits of $L^{1}$-functions under doubly stochastic transformations, Trans. Amer. Math. Soc. 117 (1965), 92–100. MR 209866, DOI 10.1090/S0002-9947-1965-0209866-5
Jesús Suárez de la Fuente, The Kalton centralizer on $L_p[0,1]$ is not strictly singular, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3447–3451. MR 3080167, DOI 10.1090/S0002-9939-2013-11599-6