Invariant subspaces of analytic perturbations

A analytic perturbations are understood here as shifts of the form $M_z + F$, where $M_z$ is the unilateral shift and $F$ is a finite rank operator on the Hardy space over the open unit disk. Here the term "a shift" refers to the multiplication operator $M_z$ on some analytic reproducing kernel Hilbert space. In this paper, first, a natural class of finite rank operators is isolated for which the corresponding perturbations are analytic, and then a complete classification of invariant subspaces of those analytic perturbations is presented. Some instructive examples and several distinctive properties (like cyclicity, essential normality, hyponormality, etc.) of analytic perturbations are also described.