Reducing Subspace Of Analytic Toeplitz Operators On N_?-Type Quotient Modules On The Torus

This thesis deals mainly with the problem of the reducing subspaces of the an-alytic Toeplitz operator S_ψ(z)on N_φ-type quotient modules on the torus.In particu-lar,we consider the cases when the symolψ(z)is z(?)(z-α_l)/(1-(?)_lz(|α_l|＞0,α_l≠α_κ((?)≠k,1≤l,κ≤N-1))or even a common finite Blaschke product.At last,we study the existence of re-ducing subspaces of S_ψ(z)from the super-isometrically dilatable operators.Chapter one is introduction.Chapter two is devoted to investigating the problem of the reducing subspaces of the analytic Tocplitz operator with symbol z^N(N≥1)on N_φ-type quotient mod-ules on the torus.The existence of its reducing subspace has been proved.What's more,a complete description has been given.Chapter three aims at proving that the Toeplitz operator with finite Blaschke product symbol S_ψ(z)on N_φ-type quotient modules on the torus has at least m non-trivial minimal reducing subspaces where m is the dimension of H²(Γ_ω)(?)φ(ω)H²(Γ_ω). Moreover,the restriction of S_ψ(z)on any of these minimal reducing subspaces is uni-tary equivalent to the Bergman shift M_z.Chapter four is concentrated at studying the existence of reducing subspaces of the analytic Toeplitz operator S_ψ(z)on N_φwith finite Blaschke product from the super-isometrically dilatable operators.