Localizations and Essential Commutant of Toeplitz Algebra on Polydisk

Usually, the norm closure of a family of operators is not equal to the C^*-algebra generated by this family of operators. But, similar with the Bergman space L²ₐ (B, dv) of the unit ball in Cⁿ, we show that the norm closure of \Tf: f L^{ (D, dv) \} on Bergman space L²ₐ (D, dv) of the ploydisk D in Cⁿ actually coincides with the Toeplitz algebra T (D). A key ingredient in the proof is the class of operators D recently introduced by Yi Wang and Jingbo Xia. In fact, as a by-product, we simultaneously proved that T (D) also coincides with D. Based on these results, we further proved that the essential commutant of Toeplitz algebra T (D) equals to \Tg: g VO₁₃₃\ + K where VO₁₃₃ is the collection of functions of vanishing oscillation on polydisk D and K denotes the collection of compact operators on L²ₐ (D, dv). On the other hand, we also prove that the essential commutant of \Tg: g VO₁₃₃\ is T (D), which implies that image of T (D) in the Calkin algebra satisfies the double commutant relation: (T (D) ) = (T (D) ) ''.