2018 年 70 巻 2 号 p. 853-875
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 𝐻2(𝔻2), the Hardy space over the bidisk, is irreducible.
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