A Gowers tree like space and the space of its bounded linear operators
Tom 190 / 2009
Studia Mathematica 190 (2009), 233-281 MSC: 46B20, 46B26. DOI: 10.4064/sm190-3-2
The famous Gowers tree space is the first example of a space not containing $c_0$, $\ell _1$ or a reflexive subspace. We present a space with a similar construction and prove that it is hereditarily indecomposable (HI) and has $\ell _2$ as a quotient space. Furthermore, we show that every bounded linear operator on it is of the form $\lambda I+W$ where $W$ is a weakly compact (hence strictly singular) operator.