N_(ψ,φ)-type Quotient Modules over the Bidisk

Let H~2■ be the Hardy space over ~the bidisk ■, and let M_(ψ,φ)=(ψ(z)-φ(w))~2] be the submodule generated by(ψ(z)-φ(w))~2, where ψ(z) and φ(w) are nonconstant inner functions.The related quotient module is denoted by ■. In this paper, we give a complete characterization for the essential normality of N_(ψ,φ). In particular, if ψ(z) = z, we simply write M_(ψ,φ)and N_(ψ,φ) as M_φ and N_φ respectively. This paper also studies compactness of evaluation operators L(0)|N_φand R(0)|N_φ, essential spectrum of compression operator S_z on N_φ, essential normality of compression operators S_z and S_w on N_φ.

Nψ,φ-type Quotient Modules over the Bidisk

Let H²(D²) be the Hardy space over the bidisk D², and let M_ψ,φ = (ψ(z) - φ(w))²] be the submodule generated by (ψ(z) - φ(w))², where ψ(z) and φ(w) are nonconstant inner functions. The related quotient module is denoted by N_ψ,φ = H²(D²) ? M_ψ,φ. In this paper, we give a complete characterization for the essential normality of N_ψ,φ. In particular, if ψ(z) = z, we simply write M_ψ,φ and N_ψ,φ as M_φ and N_φ respectively. This paper also studies compactness of evaluation operators L(0)|_Nφ and R(0)|_Nφ, essential spectrum of compression operator S_z on N_φ, essential normality of compression operators S_z and S_w on N_φ.