LAUSR

In this paper we define a graded structure induced by operators on a Hilbert space. Then we introduce several concepts which are related to the graded structure and examine some of their basic properties. A theory concerning minimal property and unitary equivalence is then developed. It allows us to obtain a complete description of V(Mzk ) on any H2(ω). It also helps us to find that a multiplication operator induced by a quasi-homogeneous polynomial must have a minimal reducing subspace. After a brief review of multiplication operator Mz+w on H2(ω, δ), we prove that the Toeplitz operator Tz+w on H 2(D2), the Hardy space over the bidisk, is irreducible.