The graded structure induced by operators on a Hilbert space

Abstract

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^*(𝑀_𝑧^𝑘) 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 𝑀_𝑧+𝑤 on 𝐻²(𝜔,𝛿), we prove that the Toeplitz operator 𝑇_𝑧+𝑤 on 𝐻²(𝔻²), the Hardy space over the bidisk, is irreducible.