Journal of the Mathematical Society of Japan

The graded structure induced by operators on a Hilbert space

Kunyu GUO and Xudi WANG

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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 $\mathcal{V}^\ast(M_{z^k})$ on any $H^2(\omega)$. 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 $H^2(\omega,\delta)$, we prove that the Toeplitz operator $T_{z+\overline{w}}$ on $H^2(\mathbb{D}^2)$, the Hardy space over the bidisk, is irreducible.

Article information

J. Math. Soc. Japan, Volume 70, Number 2 (2018), 853-875.

First available in Project Euclid: 18 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Secondary: 47C15: Operators in $C^*$- or von Neumann algebras 47A80: Tensor products of operators [See also 46M05]

unilateral weighted shifts reducing subspaces multiplication operators graded structure Toeplitz operators


GUO, Kunyu; WANG, Xudi. The graded structure induced by operators on a Hilbert space. J. Math. Soc. Japan 70 (2018), no. 2, 853--875. doi:10.2969/jmsj/07027503.

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