Maximal, minimal, and primary invariant subspaces

Let X be a complex infinite dimensional Banach space. An operator L on X is called of subcritical class, if ∑_n^∞=₁ n^-3/2 log ⁺∥Lⁿ∥ < ∞. Assume that T is an operator on X whose iterates have norms of polynomial growth. We prove that if T has a range of finite codimension and a left inverse of subcritical class, then every maximal invariant subspace of T has codimension one, and if T has a finite dimensional kernel and a right inverse of subcritical class, then every minimal invariant subspace of T is one dimensional. Using these results we obtain new information on the invariant subspace lattices of shifts and backward shifts on a wide class of Banach spaces of analytic functions on the unit disc. We also introduce the notion of primary invariant subspaces, and determine their structure for a large class of shifts.