Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

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 $\overline{\mathbb{D}}$, then such operators are of the form $T= U\oplus R$, where $U$ is isometric and $R$ is unitarily equivalent to the operator of multiplication by the variable $z$ on a de Branges-Rovnyak space $\mathcal{H}(B)$. In fact, the space $\mathcal{H}(B)$ is defined in terms of a rational operator-valued Schur function $B$. In the case when $\dim \ker T^*=1$, then $\mathcal{H}(B)$ can be taken to be a space of scalar-valued analytic functions in $\mathbb{D}$, and the function $B$ has a mate $a$ defined by $|B|^2+|a|^2=1$ a.e. on $\partial \mathbb{D}$. We show the mate $a$ of a rational $B$ is of the form $a(z)=a(0)\frac{p(z)}{q(z)}$, 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 $\mathcal{H}(B)$ by use of what we call the local Dirichlet integral of order $m$ at the point $w\in \partial \mathbb{D}$.