Invariant subspaces of analytic perturbations

Analytic perturbations are understood here as shifts of the form M z + F Mᵦ + F, where M z Mᵦ is the unilateral shift and F 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 Mᵦ 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.