대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제54권4호
  • /
  • Pages.1443-1455
  • /
  • 2017
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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REDUCING SUBSPACES OF A CLASS OF MULTIPLICATION OPERATORS

  • Liu, Bin (School of Mathematical Sciences Ocean University of China) ;
  • Shi, Yanyue (School of Mathematical Sciences Ocean University of China)
  • 투고 : 2016.07.25
  • 심사 : 2016.11.29
  • 발행 : 2017.07.31
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초록

Let $M_{z^N}(N{\in}{\mathbb{Z}}^d_+)$ be a bounded multiplication operator on a class of Hilbert spaces with orthogonal basis $\{z^n:n{\in}{\mathbb{Z}}^d_+\}$. In this paper, we prove that each reducing subspace of $M_{z^N}$ is the direct sum of some minimal reducing subspaces. For the case that d = 2, we find all the minimal reducing subspaces of $M_{z^N}$ ($N=(N_1,N_2)$, $N_1{\neq}N_2$) on weighted Bergman space $A^2_{\alpha}({\mathbb{B}}_2)$(${\alpha}$ > -1) and Hardy space $H^2({\mathbb{B}}_2)$, and characterize the structure of ${\mathcal{V}}^{\ast}(z^N)$, the commutant algebra of the von Neumann algebra generated by $M_{z^N}$.

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과제정보

연구 과제 주관 기관 : NSFC

참고문헌

  1. M. Albaseer, Y. Lu, and Y. Shi, Reducing subspaces for a class of Toeplitz Operators on the Bergman space of the bidisk, Bull. Korean Math. Soc. 52 (2015), no. 5, 1649-1660. https://doi.org/10.4134/BKMS.2015.52.5.1649
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  3. K. Guo and H. Huang, Multiplication operators on the Bergman space, Lecture Notes in Mathematics, 2145, Springer, 2015.
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  7. X. Wang, H. Dan, and H. Huang, Reducing subspaces of multiplication operators with the symbol ${\alpha}z^k$ + ${\beta}w^l$ on $L\frac{2}{{\alpha}}({\mathbb{D}}^2)$, Sci. China Math. 58 (2015), no. 10, 2167-2180.