arXiv subjects

J. M. F. Castillo

Publications and source records attributed to J. M. F. Castillo.

3 recordsLinked to original sources

arXiv2026

A twisted Hilbert space not isomorphic to its dual

We show: 1) The existence of the first twisted Hilbert space that is not isomorphic to its dual; this solves a problem posed by Cabello in [Nonlinear centralizers in homology, Math. Ann. 358 (2014), no. 3-4, 779-798]. 2) The existence of a large coneable family of relatively incomparable such examples, improving the coneable family obtained in [W.H. Corrêa, S. Dantas, D.L. Rodríguez-Vidanes, Twisted Hilbert spaces defined by Lipschitz embeddings, Israel J. of Mathematics, to appear]. 3) The existence of quasilinear maps between Hilbert spaces not isomorphic to Kalton centralizers; which solves another question of Cabello. 4) The existence of a large family of mutually incomparable elements in the ordered set of twisted Hilbert exact sequences. This complements earlier results in [J.M.F. Castillo, W. Cuellar, V. Ferenczi, Y. Moreno, Complex structures on twisted Hilbert spaces, Israel J. Math. 222 (2017) 787-814] -- where it was proved that the ordered set did not have a first element -- and [F. Cabello Sánchez, J.M.F. Castillo, W.H.G. Corrêa, V. Ferenczi, R. García, On the $Ext^2$-problem in Hilbert spaces, J. Funct. Anal. 280 (2021) 108863] -- where two incomparable elements were obtained.

math.FA↗

arXiv2019

Differential processes generated by two interpolators

We study couples of interpolators, the differentials they generate and their associated commutator theorems. An essential part of our analysis is the study of the intrinsic symmetries of the process. Since we work without any compatibility or categorical assumption, our results are flexible enough to generalize most known results for commutators or translation operators, in particular those of Cwikel, Kalton, Milman, Rochberg \cite{ckmr} for differential methods and those of Carro, Cerdà and Soria \cite{caceso} for compatible interpolators. We also generalize stability and singularity results in \cite{cfg,ccfg,correa} from the complex method to general differential methods and obtain new incomparability results.

math.FA↗

arXiv2011

Local complementation and the extension of bilinear mappings

We study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if $X$ is not a Hilbert space then one may find a subspace of $X$ for which there is no Aron-Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given $X$ forces such $X$ to contain no uniform copies of $\ell_p^n$ for $p\in[1,2)$. In particular, $X$ must have type $2-ε$ for every $ε>0$. Also, we show that the bilinear version of the Lindenstrauss-Pełczyński and Johnson-Zippin theorems fail. We will then consider the notion of locally $α$-complemented subspace for a reasonable tensor norm $α$, and study the connections between $α$-local complementation and the extendability of $α^*$ -integral operators.

math.FA↗