An operator model in the annulus

For an invertible linear operator $T$ on a Hilbert space $H$, put $$\alpha(T^*,T) := -T^{*2}T^2 + (1+r^2) T^* T - r^2 I,$$ where $I$ stands for the identity operator on $H$ and $r\in (0,1)$; this expression comes from applying Agler's hereditary functional calculus to the polynomial $\alpha(t)=(1-t) (t-r^2)$. We give a concrete unitarily equivalent functional model for operators satisfying $\alpha(T^*,T)\ge0$. In particular, we prove that the closed annulus $r\le |z|\le 1$ is a complete $K$-spectral set for $T$. We explain the relation of the model with the Sz.-Nagy--Foias one and with the observability gramian and discuss the relationship of this class with other operator classes related to the annulus.