quasihomogeneous symbols
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2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Qi Wu ◽  
Chuntao Qin ◽  
Yong Chen ◽  
Yile Zhao

We study (semi)commutativity of small Hankel operators with separately quasihomogeneous symbols on the pluriharmonic Bergman space of the unit ball. Some product problems are also concerned.


2014 ◽  
Vol 90 (3) ◽  
pp. 494-503 ◽  
Author(s):  
XING-TANG DONG ◽  
CONGWEN LIU ◽  
ZE-HUA ZHOU

AbstractIn this paper, we completely determine the commutativity of two Toeplitz operators on the harmonic Bergman space with integrable quasihomogeneous symbols, one of which is of the form $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}e^{ik\theta }r^{\, {m}}$. As an application, the problem of when their product is again a Toeplitz operator is solved. In particular, Toeplitz operators with bounded symbols on the harmonic Bergman space commute with $T_{e^{ik\theta }r^{\, {m}}}$ only in trivial cases, which appears quite different from results on analytic Bergman space in Čučković and Rao [‘Mellin transform, monomial symbols, and commuting Toeplitz operators’, J. Funct. Anal.154 (1998), 195–214].


2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jingyu Yang ◽  
Liu Liu ◽  
Yufeng Lu

We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space. We first give the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator and discuss the zero-product problem for several Toeplitz operators with radial symbols. Next, we study the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols. Finally, we also investigate finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols.


2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Jun Yang

We characterize the commuting Toeplitz operator and Hankel operator with quasihomogeneous symbols. Also, we use it to show the necessary and sufficient conditions for commuting Toeplitz operator and Hankel operator with ordinary functions.


2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Bo Zhang ◽  
Yufeng Lu

We consider when the product of two Toeplitz operators with some quasihomogeneous symbols on the Bergman space of the unit ball equals a Toeplitz operator with quasihomogeneous symbols. We also characterize finite-rank semicommutators or commutators of two Toeplitz operators with quasihomogeneous symbols.


2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Bo Zhang ◽  
Yanyue Shi ◽  
Yufeng Lu

We discuss some algebraic properties of Toeplitz operators on the Bergman space of the polydiskDn. Firstly, we introduce Toeplitz operators with quasihomogeneous symbols and property (P). Secondly, we study commutativity of certain quasihomogeneous Toeplitz operators and commutators of diagonal Toeplitz operators. Thirdly, we discuss finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols. Finally, we solve the finite rank product problem for Toeplitz operators on the polydisk.


2005 ◽  
Vol 85 (3) ◽  
pp. 248-257 ◽  
Author(s):  
Issam Louhichi ◽  
Lova Zakariasy

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