A note on a Halmos problem

Abstract

We address the existence of non-trivial closed invariant subspaces of operators $T$ on Banach spaces whenever their square $T^2$ have or, more generally, whether there exists a polynomial $p$ with $\text{deg}\left(p\right)⩾2$ such that the lattice of invariant subspaces of $p\left(T\right)$ is non-trivial. In the Hilbert space setting, the $T^2$-problem was posed by Halmos in the seventies and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to the Invariant Subspace Problem.