Hankel算子的乘积与有限秩算子
The Product of Hankel Operators and the Finite Rank Operators
设f,g,u ∈ ∩q>1 Hq,Hf,Hg,Hu均为通常的单位圆盘上的Hardy空间H2到H2上的Hankel算子.本文完全刻画了Hardy空间上的三个Hankel算子的乘积HfHgHu是有限秩的充要条件,并给出了两个不平凡的例子.而且,我们利用本文的主要结果刻画了模型空间上有限秩的截断Toeplitz算子.
Suppose f, g, u ∈ ∩q>1 Hq, Hf, Hg, Hu are Hankel operators from the usual Hardy space of unit disk H2 to H2. In this paper, we completely characterize when the product of three Hankel operators i>HfHgHu on Hardy space has finite rank property, and we also give two nontrivial examples. Moreover, we describe the finite rank property of truncated Toeplitz operators defined on the model space.
Hardy空间 / Hankel算子 / 截断Toeplitz算子 {{custom_keyword}} /
Hardy spaces / Hankel operators / truncated Toeplitz operators {{custom_keyword}} /
[1] Axler S., Chang S. Y., Sarason D., Products of Toeplitz operators, Integr. Equ. Oper. Theory, 1978, 1:285-309.
[2] Aleman A., Vukoti? D., Zero products of Toeplitz operators, Duke Math. J., 2009, 148:373-403.
[3] Baranov A., Chalendar I., Fricain E., et al., Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators, J. Funct. Anal., 2010, 259:2673-2701.
[4] Bessonov R., Truncated Toeplitz operators of finite rank, Proc. Amer. Math. Soc., 2014, 142(4):1301-1313.
[5] Brown A., Halmos P., Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 1964, 213:89-102.
[6] Conway J., A Course in Functional Analysis, 2nd Edition, Springer-Verlag, New York, 1994.
[7] Ding X. H., The finite rank perturbations of the product of Hankel and Toeplitz operators, J. Math. Anal. Appl., 2008, 337(1):726-738.
[8] Ding X. H., Sang Y. Q., Two questions on products of Hankel operators (in Chinese), Sci. Sin. Math., doi:10.1360/NO12019-00017.
[9] Ding X. H., Zheng D. C., Finite rank commutator of Toeplitz operators or Hankel operators, Houston J. Math., 2008, 34(4):1099-1120.
[10] Douglas R., Banach Algebra Techniques in Operator Theory, 2nd Edition, Springer-Verlag, New York, 1998.
[11] Gu C. X., Finite rank products of four Hankel operators, Houston J. Math., 1999, 25(3):543-561.
[12] Gu C. X., Products of several Toeplitz operators, J. Funct. Anal., 2000, 171:483-527.
[13] Guo K. Y., A problem on products of Toeplitz operators, Proc. Amer. Math. Soc., 1996, 124:869-871.
[14] Peller V., Hankel Operators and Their Applications, Springer-Verlag, New York, 2003.
[15] Sarason D., Algebraic properties of truncated Toeplitz operators, Oper. Matrices, 2007, 1(4):491-526.
[16] Xia D. X., Zheng D. C., Products of Hankel operators, Integr. Equ. Oper. Theory, 1997, 29:339-363.
[17] Zhu K. H., Operator Theory in Function Spaces, 2nd Edition, Amer. Math. Soc., Providence, RI, 2007.
国家自然科学基金资助项目(11871122);重庆市自然科学基金(cstc2018jcyjAX0595,cstc2020jcyj-msxmX0318);重庆工商大学基金(2053010)
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