Complex structures on twisted Hilbert spaces

Pacific Journal of Mathematics, 2015

We show that Rochberg's generalized interpolation spaces ᐄ (n) arising from analytic families of Banach spaces form exact sequences 0 → ᐄ (n) → ᐄ (n+k) → ᐄ (k) → 0. We study some structural properties of those sequences; in particular, we show that nontriviality, having strictly singular quotient map, or having strictly cosingular embedding depend only on the basic case n = k = 1. If we focus on the case of Hilbert spaces obtained from the interpolation scale of p spaces, then ᐄ (2) becomes the well-known Kalton-Peck space Z 2 ; we then show that ᐄ (n) is (or embeds in, or is a quotient of) a twisted Hilbert space only if n = 1, 2, which solves a problem posed by David Yost; and that it does not contain 2 complemented unless n = 1. We construct another nontrivial twisted sum of Z 2 with itself that contains 2 complemented.

2014

We show that Rochberg's generalizared interpolation spaces X (n) arising from analytic families of Banach spaces form exact sequences 0 → X (n) → X (n+k) → X (k) → 0. We study some structural properties of those sequences; in particular, we show that nontriviality, having strictly singular quotient map, or having strictly cosingular embedding depend only on the basic case n = k = 1. If we focus on the case of Hilbert spaces obtained from the interpolation scale of ℓ p spaces, then X (2) becomes the well-known Kalton-Peck Z 2 space; we then show that X (n) is (or embeds in, or is a quotient of) a twisted Hilbert space only if n = 1, 2, which solves a problem posed by David Yost; and that it does not contain ℓ 2 complemented unless n = 1. We construct another nontrivial twisted sum of Z 2 with itself that contains ℓ 2 complemented.

Journal of the Institute of Mathematics of Jussieu, 2003

We show that a twisted Hilbert space with an unconditional basis is isomorphic to a Hilbert space.

Integral Equations and Operator Theory

We study those operators on a Hilbert space that can be lifted or extended to any twisted Hilbert space. We prove that these form an ideal of operators which contains all the Schatten classes. We characterize those multiplication operators on $$\ell _p$$ ℓ p that are liftable/extensible through centralizers.

We present new methods to obtain singular twisted sums X ⊕Ω X (i.e., exact sequences 0 → X → X ⊕Ω X → X → 0 in which the quotient map is strictly singular), in which X is the interpolation space arising from a complex interpolation scheme and Ω is the induced centralizer.

Journal of Mathematical Analysis and Applications, 2002

In this paper, we study the structure of quasi-invariant subspaces of analytic Hilbert spaces over the complex plane. We especially investigate when two quasiinvariant subspaces are similar or unitarily equivalent for an analytic Hilbert space over the complex plane.  2002 Elsevier Science (USA)

Journal of Geometry and Physics, 2006

The twistor construction is applied for obtaining examples of generalized complex structures (in the sense of N. Hitchin) that are not induced by a complex or a symplectic structure.

Journal of Geometry and Physics, 2014

Annales de l’institut Fourier, 2011

<http://aif.cedram.org/item?id=AIF_2011__61_6_2219_0> © Association des Annales de l'institut Fourier, 2011, tous droits réservés. L'accès aux articles de la revue « Annales de l'institut Fourier » (http://aif.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://aif.cedram.org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

We study complex structures in real two-dimensional commutative algebras and show their connection with homotopy properties of the multiplications. An application to the Riccati equation in rank three algebras is also discussed.