Abstract: It is proved that there are some Hypercyclic and Supercyclic operators in the class of co-analytic Toeplitz operators in Hardy space and Bergman space. Applying spectrum and index theorys, we also prove that Toeplitz operator T_φ belongs to H_c(H) or S_c(H) when the symbol φ satisfies certain conditions.

Key words: analytic Toeplitz operator, co-analytic Toeplitz opera tor, Hypercyclic operator, Supercyclic operator