Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

Abstract

We investigate expansive Hilbert space operators T that are finite rank perturbations of isometric operators. If the spectrum of T is contained in the closed unit disc D‾, then such operators are of the form T=U⊕R, where U is isometric and R is unitarily equivalent to the operator of multiplication by the variable z on a de Branges-Rovnyak space H(B). In fact, the space H(B) is defined in terms of a rational operator-valued Schur function B. In the case when dim⁡ker⁡T⁎=1, then H(B) can be taken to be a space of scalar-valued analytic functions in D, and the function B has a mate a defined by |B|2+|a|2=1 a.e. on ∂D. We show the mate a of a rational B is of the form a(z)=a(0)p(z)q(z), where p and q are appropriately derived from the characteristic polynomials of two associated operators. If T is a 2m-isometric expansive operator, then all zeros of p lie in the unit circle, and we completely describe the spaces H(B) by use of what we call the local Dirichlet integral of order m at the point w∈∂D.