Invariant subspaces for operator semigroups with commutators of rank at most one

TL;DR

This paper proves that certain operator semigroups with rank-one commutator constraints on a complex Banach space have non-trivial invariant subspaces unless they are commutative, and shows they are triangularizable if polynomially compact.

Contribution

It establishes the existence of invariant subspaces for non-commutative semigroups with rank-one commutator conditions and extends triangularization results to polynomially compact operators.

Findings

Semigroups with rank ≤ 1 commutators have invariant subspaces if non-commutative.

Such semigroups are triangularizable if they are polynomially compact.

Generalizes previous results on invariant subspaces and triangularization.

Abstract

Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of AB - BA is at most 1 for all pairs {A,B} in S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997), 443-456] and [G. Cigler, R. Drnov\v{s}ek, D. Kokol-Bukov\v{s}ek, T. Laffey, M. Omladi\v{c}, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998), 452-465].