Algebraic characterizations of paired operators on model spaces

We study paired multiplication operators associated with orthogonal decompositions of $L^2(\mathbb{T})$ arising from Hardy spaces and Sz.-Nagy–Foias model spaces. For a non-constant inner function $\theta$, we characterize ${\mathcal{K}}_{\theta }$-paired and triply paired operators by finite-rank commutator identities with the bilateral shift $M_z$. We also apply these ideas to sums of truncated Toeplitz and truncated Hankel operators on ${\mathcal{K}}_{\theta }$. Using the displacement characterizations of Sarason and Gu–Ma, we derive a second-order finite-rank displacement formula for such sums and describe the ambiguity of the decomposition in terms of ${\mathcal{I}}_{\theta }={\mathcal{T}}_{\theta }\cap {\mathcal{H}}_{\theta }$. In the finite-dimensional case $\theta (z)=z^n$, the framework recovers the classical Toeplitz-plus-Hankel matrix recurrence of Bevilacqua–Bonanni–Bozzo.