An operator model in the annulus

For an invertible linear operator T on a Hilbert space H, put \[ α(T^*,T) := -T*2T^2 + (1+r^2) T^* T - r^2 I, \] where I stands for the identity operator on H and r∈ (0,1); this expression comes from applying Agler's hereditary functional calculus to the polynomial α(t)=(1-t) (t-r²). We give a concrete unitarily equivalent functional model for operators satisfying α(T^*,T)≥0. In particular, we prove that the closed annulus r≤ |z|≤ 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.