On almost-invariant subspaces and approximate commutation (Q1939357)

Given a bounded linear operator \(T\) on a Banach space \(X\), a closed subspace \(Y\subset X\) is called almost-invariant for \(T\) if there exists a finite-dimensional subspace \(M\subset X\) such that \(T(Y)\subset Y+M\). Since finite dimensional (or co-dimensional) subspaces are almost-invariant for every operator, the authors' interest focuses on half-spaces, i.e., subspaces of infinite dimension and co-dimension. The paper under review studies the existence of common almost-invariant half-spaces for algebras of operators. For instance, it is proved that, if \(\mathcal A\subset \mathcal B(X)\) is a norm-closed algebra of operators on a Banach space \(X\), which admits a common almost-invariant half-space which is complemented in \(X\), then \(\mathcal A\) has a common invariant subspace. The authors also study the case of maximal abelian self-adjoint subalgebras in \(\mathcal B (H)\) for a Hilbert space \(H\) and prove, among other things, the following result: if the range of every projection in a maximal abelian self-adjoint algebra \(\mathcal D\) is almost-invariant for an operator \(T\in\mathcal B(H)\), then we can write \(T=D+F\) with \(D\in\mathcal D\) and \(F\) of finite-rank. Note that the second author and \textit{A. Tcaciuc} have subsequently proved that every operator on a separable, reflexive Banach space has almost-invariant half-spaces [J. Funct. Anal. 265, No.~2, 257--265 (2013; Zbl 1300.47015)].