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 H 2 (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 H 2 (ω, δ), we prove that the Toeplitz operator T z+w on H 2 (D 2 ), the Hardy space over the bidisk, is irreducible.