Strictly singular, non-compact operators exist on the space of Gowers and Maurey

Subsequential minimality in Gowers and Maurey spaces

We define block sequences $(x_n)$ in every block subspace of a variant of the space of Gowers and Maurey so that the map $x_{2n-1}\mapsto x_{2n} $ extends to an isomorphism. This implies the existence of a subsequentially minimal HI space, which solves a question in \cite{FR}.

A uniformly convex hereditarily indecomposable Banach space

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 …

Spectral-free methods in the theory of hereditarily indecomposable Banach spaces

We give new and simple proofs of some classical properties of hereditarily indecomposable Banach spaces, including the result by W. T. Gowers and B. Maurey that a hereditarily indecomposable Banach space cannot be isomorphic to a proper subspace of itself. These proofs do not make use of spectral theory and therefore, they work in real spaces as well as in complex spaces. We use our method to prove some new results. For example, we give a quantitative version of the latter result by Gowers and Maurey and deduce that Banach spaces that are isometric to all of their subspaces should have an unco…

一个具有无限维自反商代数 L(X)/SS(X) 的 Banach 空间

A Banach space with an infinite dimensional reflexive quotient algebra L(X)/SS(X)

构造 Gowers-Maurey 型 Banach 空间 X，使有界算子代数 L(X) 为无限维自反空间与严格奇异算子理想的直和，从而商代数 L(X)/SS(X) 为无限维自反空间。

Some results about the Schroeder-Bernstein Property for separable Banach spaces

We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder-Bernstein Index of any of these spaces is $2^{\aleph_0}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of V. Ferenczi and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder-Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.

A Banach space whose algebra of operators is Dedekind-finite but it does not have stable rank one

In this note we examine the connection between the stable rank one and Dedekind-finite property of the algebra of operators on a Banach space $X$. We show that for the indecomposable but not hereditarily indecomposable Banach space $X_{\infty}$ constructed by Tarbard (Ph.D. Thesis, University of Oxford, 2013), the algebra of operators $B(X_{\infty})$ is Dedekind-finite but does not have stable rank one. While this sheds some light on the Banach space structure of $X_{\infty}$ itself, we observe that the indecomposable but not hereditarily indecomposable Banach space constructed by Gowers and M…