arXiv subjects
Publications and source records attributed to Valentin Ferenczi.
We consider a normalized basis in a Banach space with the following property: any normalized block sequence of the basis has a subsequence equivalent to the basis. We show that under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. We also address an analogous problem for spreading models.
If a Banach space has an unconditional basis it either contains a continuum of non isomorphic subspaces or is isomorphic to its square and hyperplanes and satisfies other regularity properties. An HI Banach space contains a continuum of non isomorphic subspaces.
A Banach space $X$ is said to be Q.H.I. if eve\-ry infinite dimensional quo\-tient spa\-ce of $X$ is H.I.: that is, a space is Q.H.I. if the H.I. property is not only stable passing to subspaces, but also passing to quotients and to the dual. We show that Gowers-Maurey's space is Q.H.I.; then we provide an example of a reflexive H.I. space ${\cal X}$ whose dual is not H.I., from which it follows that $\cal X$ is not Q.H.I.
A space $X$ is said to be hereditarily indecomposable if no two (infinite dimensional) subspaces of $X$ are in a direct sum. In this paper, we show that if $X$ is a complex hereditarily indecomposable Banach space, then every operator from a subspace $Y$ of $X$ to $X$ is of the form $λI + S$, where $I$ is the inclusion map and $S$ is strictly singular.
A {\em hereditarily indecomposable (or H.I.)} Banach space is an infinite dimensional Banach space such that no subspace can be written as the topological sum of two infinite dimensional subspaces. As an easy consequence, no such space can contain an unconditional basic sequence. This notion was first introduced in 1993 by T.Gowers and B.Maurey, who constructed the first known example of a hereditarily indecomposable space. Gowers-Maurey space is reflexive, however it is not uniformly convex. In this article, we provide an example of a uniformly convex hereditarily indecomposable space, using similar methods as Gowers and Maurey, and using the theory of complex interpolation for a family of Banach spaces of Coifman, Cwikel, Rochberg, Sagher and Weiss.