A Poincaré-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem

Abstract

This paper presents a local nonlinear Beurling-Lax-Halmos type representation for a shift-invariant germ of a complex-analytic manifold embedded in a separable Hilbert space. We represent the manifold locally (near 0) by an asymptotic series (which commutes with the shift) and we discuss the possible convergence of this series. The techniques that we employ are heavily influenced by the classical Poincaré-Dulac theory in nonlinear ordinary differential equations, combined with the commutant lifting theorem.