Local commutants and ultrainvariant subspaces

TL;DR

This paper investigates the structure of local commutants of operators on Banach spaces, introducing ultrainvariant subspaces as those for which the local commutant forms an algebra, and explores their properties across different operator types.

Contribution

It characterizes when local commutants form an algebra via ultrainvariant subspaces and demonstrates their existence for various classes of operators.

Findings

For normal operators, all hyperinvariant subspaces are ultrainvariant.

Ultrainvariant subspaces can be strictly smaller than hyperinvariant subspaces for nilpotent operators.

The paper provides conditions under which local commutants are algebras.

Abstract

For an operator $A$ on a complex Banach space $X$ and a closed subspace $M\subseteq X$, the local commutant of $A$ at $M$ is the set $C(A;M)$ of all operators $T$ on $X$ such that $TAx=ATx$ for every $x\in M$. It is clear that $ C(A;M)$ is a closed linear space of operators, however it is not an algebra, in general. For a given $A$, we show that $C(A;M)$ is an algebra if and only if the largest subspace $M_A$ such that $C(A;M)=C(A;M_A)$ is invariant for every operator in $C(A;M)$. We say that these are ultrainvariant subspaces of $A$. For several types of operators we prove that there exist non-trivial ultrainvariant subspaces. For a normal operator on a Hilbert space, every hyperinvariant subspace is ultrainvariant. On the other hand, the lattice of all ultrainvariant subspaces of a non-zero nilpotent operator can be strictly smaller than the lattice of all hyperinvariant subspaces.