Dirichlet空间上一类积分算子的约化子空间

数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 595-602.

Dirichlet空间上一类积分算子的约化子空间

张核心()

^{河北师范大学数学科学学院石家庄 050024}

收稿日期:2020-03-20 出版日期:2021-06-26 发布日期:2021-06-09
作者简介:张核心, E-mail: corehebtu@126.com
基金资助:
河北师范大学研究生创新资助项目(CXZZSS2020052)

On Reducing Subspaces of a Class of Integral Operator on Dirichlet Spaces

Hexin Zhang()

^{School of Mathematical and Sciences, Hebei Normal University, Shijiazhuang 050024}

Received:2020-03-20 Online:2021-06-26 Published:2021-06-09
Supported by:
the Postgraduate Innovative Funding Project of Hebei Normal University(CXZZSS2020052)

摘要/Abstract

摘要：

刻画了n移位算子加带特定权的Volterra算子T₁在Dirichlet空间上的相似性，利用算子理论技巧证明了$ T_{1}$在Dirichlet空间上的作用和乘法算子$M_{p} $在空间$S（{\Bbb D}） $上的作用相似，进一步证明了当$p（z）=z^{n} $时相应的算子$T_{2} $有$ 2^{n}$个约化子空间.

关键词: Dirichlet空间, Volterra算子, n移位算子, 约化子空间

Abstract:

We characterize the similarity of $n$-shift plus certain weighted Volterra operator denoted by $T_{1}$ on the Dirichlet space ${\cal D}$, and prove that $T_{1}$ acting on ${\cal D}$ is similar to the multiplication operator $M_{p}$ acting on $S({\Bbb D})$ by using some techniques in operator theory. Furthermore, we consider the case of $p(z)=z^{n}$ corresponding to the operator $T_{2}$, and prove that the operator $T_{2}$ has exactly $2^{n}$ reducing subspaces.

Key words: Dirichlet space, Volterra operator, n-Shift operator, Reducing subspaces

中图分类号:

O177.2

张核心. Dirichlet空间上一类积分算子的约化子空间[J]. 数学物理学报, 2021, 41(3): 595-602.

Hexin Zhang. On Reducing Subspaces of a Class of Integral Operator on Dirichlet Spaces[J]. Acta mathematica scientia,Series A, 2021, 41(3): 595-602.

参考文献 14

1	Albaseer M , Lu Y F , Shi Y Y . Reducing subspaces for a class of Toeplitz operators on the Bergman space of the bidisk. Bull Korean Math Soc, 2015, 52 (5): 1649- 1660 doi: 10.4134/BKMS.2015.52.5.1649
2	Beurling A . On two problems concerning linear transformations in Hilbert space. Acta Math, 1949, 81, 239- 255 doi: 10.1007/BF02395019
3	Douglas R G , Kim Y S . Reducing subspaces on the annulus. Integral Equations Operator Theory, 2011, 70 (1): 1- 15 doi: 10.1007/s00020-011-1874-3
4	Guo K Y , Huang H S . On multiplication operators on the Bergman space: similarity, unitary equivalence and reducing subspaces. J Operator Theory, 2011, 65 (2): 355- 378
5	Guo K Y , Wang X D . Reducing subspaces of tensor products of weighted shifts. Sci China Math, 2016, 59 (4): 715- 730 doi: 10.1007/s11425-015-5089-y
6	Luo S B . Reducing subspaces of multiplication operators on the Dirichlet spaces. Integral Equations Operator Theory, 2016, 85 (4): 539- 554 doi: 10.1007/s00020-016-2295-0
7	Lu Y F , Zhou X Y . Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk. J Math Soc Japan, 2010, 62 (3): 745- 765
8	Shi Y Y , Lu Y F . Reducing subspaces for Toeplitz operators on the polydisk. Bull Korean Math Soc, 2013, 50 (2): 687- 696 doi: 10.4134/BKMS.2013.50.2.687
9	Stessin M , Zhu K H . Reducing subspaces of weighted shift operators. Proc Amer Math Soc, 2002, 130 (9): 2631- 2639 doi: 10.1090/S0002-9939-02-06382-7
10	Zhao L K . Reducing subspaces for a class of multiplication operators on the Dirichlet space. Proc Amer Math Soc, 2009, 137 (9): 3091- 3097 doi: 10.1090/S0002-9939-09-09859-1
11	Jiang C L , Li Y C . The commutant and similarity invariant of analytic Toeplitz operators on Bergman space. Sci China Math Ser A, 2007, 50 (5): 651- 664 doi: 10.1007/s11425-007-0027-2
12	Jiang C L , Zheng D C . Similarity of analytic Toeplitz operators on the Bergman spaces. J Funct Anal, 2010, 258 (9): 2961- 2982 doi: 10.1016/j.jfa.2009.09.011
13	Li Y C , Chen H , Lan W H . On similarity and reducing subspaces of the n-shift plus certain weighted Volterra operator. J Funct Spaces, 2017, doi: 10.1155/2017/8370139
14	Douglas R G . Banach Algebra Techniques in Operator Theory. New York: Academic Press, 1972

[1]	夏锦, 王晓峰, 曹广福. 调和Dirichlet空间D_h¹上有界、紧与Fredholm的Toeplitz算子[J]. 数学物理学报, 2015, 35(5): 956-969.
[2]	陈建军, 王晓峰. 圆环上Dirichlet空间中的Toeplitz算子及其代数性质[J]. 数学物理学报, 2014, 34(4): 890-904.
[3]	夏锦, 王晓峰, 曹广福. Dirichlet空间上Toeplitz算子的一些性质[J]. 数学物理学报, 2012, 32(2): 395-403.
[4]	夏锦, 王晓峰, 曹广福. Dirichlet空间上的紧算子[J]. 数学物理学报, 2009, 29(5): 1196-1205.