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A twisted Hilbert space not isomorphic to its dual

J. M. F. Castillo, W. H. G. Corrêa

Abstract

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.

A twisted Hilbert space not isomorphic to its dual

Abstract

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 -problem in Hilbert spaces, J. Funct. Anal. 280 (2021) 108863] -- where two incomparable elements were obtained.
Paper Structure (13 sections, 13 theorems, 30 equations)

This paper contains 13 sections, 13 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exact sequences of Banach spaces
  4. Quasilinear maps and centralizers
  5. Twisted Hilbert spaces
  6. Two problems of Cabello
  7. Problem 1
  8. The family $Z(\varphi, \psi)$
  9. Problem 2.
  10. Incomparable elements in $\operatorname{Ext}(\ell_2, \ell_2)$
  11. Further considerations and unanswered questions
  12. Twisted Hilbert sequences not isomorphically equivalent to its dual
  13. About being equivalent to its dual

Key Result

Lemma 2.1

An operator $T: \ell_2\longrightarrow X$ is almost-compact if and only if for every infinite $N\subset \mathbb{N}$ there exists an infinite subset $M\subset N$ such that the restriction $T|_{\ell_2(M)}$ is compact.

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • proof : Proof of the Claim
  • ...and 13 more