A note on a Halmos problem

TL;DR

This paper explores the existence of non-trivial invariant subspaces for operators on Banach and Hilbert spaces, focusing on the implications of the operator's polynomial functions, addressing a longstanding problem posed by Halmos.

Contribution

It investigates the relationship between the invariant subspaces of an operator and those of its polynomial functions, extending the classical Halmos problem to broader settings.

Findings

Addresses the Halmos problem in the context of Banach and Hilbert spaces.

Examines the invariant subspace structure of polynomial functions of operators.

Provides insights into the equivalence between the T^2-problem and the Invariant Subspace 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 $\mbox{deg}(p)\geq 2$ such that the lattice of invariant subspaces of $p(T)$ 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 \emph{Invariant Subspace Problem}.