Subsymmetric weak$^*$ Schauder bases and factorization of the identity
Volume 248 / 2019
Studia Mathematica 248 (2019), 295-319
MSC: 46B25, 46B26.
DOI: 10.4064/sm180404-29-9
Published online: 22 March 2019
Abstract
We provide conditions on a dual Banach space $X^*$ with a subsymmetric weak$^*$ Schauder basis which allow us to ensure that for any bounded operator $T \colon X^*\to X^*$, either $T(X^*)$ or $({\rm Id}_{X^*}-T)(X^*)$ contains a subspace that is isomorphic to $X^*$ and complemented in $X^*$. Under the same conditions on $X^*$, we prove that $\ell ^p(X^*)$, $1\leq p \leq \infty $, is primary. Moreover, we show that these conditions are satisfied by a wide range of Orlicz and Lorentz sequence spaces.
