Contractive commutants and invariant subspaces

Abstract:

Let $T$ be a bounded operator on a Banach space $\mathfrak {X}$ and let $K$ be a nonzero compact operator. In [1] and [4] it is shown that if $\lambda$ is a complex number and if $TK=\lambda KT$, then $T$ has a hyperinvariant subspace. In [1], S. Brown goes on to show that if $\mathfrak {X}$ is reflexive and if $TK = \lambda KT$ and $TB = \mu BT$ for some $\lambda$, $\mu$ with $\left | \lambda \right | \ne 1$ and $(1 - \left | \mu \right |)/(1 - \left | \lambda \right |) \geqslant 0$, then $B$ has an invariant subspace. Below we extend both these results by showing that the entire class of operators satisfying the above conditions on $B$ has an invariant subspace.