Reducing Subspaces of Toeplitz Operators on $N_ϕ$-Type Quotient Modules on the Torus (Communications in Mathematical Research, 2021)
In this paper, we prove that the Toeplitz operator with finite Blaschkeproduct symbol $S_{ψ(z)}$ on $N_ϕ$ has at least $m$ non-trivial minimal reducing subspaces,where $m$ is the dimension of $H^2(Γ_ω) ⊖ ϕ(ω)H^2(Γ_ω)$. Moreover, the restrictionof $S_{ψ(z)}$ on any of these minimal reducing subspaces is unitary equivalent to theBergman shift $M_z$.
Keywords: module; $N_ϕ$-type quotient module; the analytic Toeplitz operator; reducing subspace; finite Blaschke product