调和Dirichlet空间上小Hankel算子的乘积(英文)论文
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AbstractIn this paper， we study all Hankel operators with harmonic symbols on the harmonic Dirichlet space， completely characterize the condition for the cummutativity and zero-product of all Hankel operators.
Key wordsHarmonic Dirichlet space； Small Hankel operator； Commutativity； Zero product CLC numberO 17
E
Recently， products of Toeplitz operators and Hankel operators on the Dirich-
let space he been studied intensively[2？9]. As has implied[10？15]， the theory of Toeplitz operators on harmonic function space is quite different from the theory on analytic function space. For the operators on harmonic Dirichlet space， （semi-） commutativity[15]and zero-product[16]of Toeplitz operators on Dhwith symbols in M he been completely characterize. In this paper， for symbols in M， we consider the problem when two all Hankel operators is commutativity. By using the matrix representation of all Hankel operator on harmonic Dirichlet space， we give a complete characterization of such operators to be commutativity. By using a similar method， the zero-product of all Hankel operators is also characterized.
2Preliminaries
Recall that U is the operator on S defined by （Uf）（z） = f（ˉz）， f∈S， and U is an unitary on S， U？= U = U？1[1]. For simplicity， for f∈S， denote？f as Uf， i.e.，？f（z） = f（ˉz）.
Let P be the orthogonal projection from S onto D and P1be the orthogonal projection from S ontoˉD， then P1= UPU[1]. Similarly， for the orthogonal projection Q， we he the following result.
As the computation in Theorem 1， by equations above， we obtain the following equations.
Compare equations （14），（18）， （19） and equations （16）， （19）， we he the following equations （a′） and （b′） respectively. Compare equation （15），（18） and equations（17），（18）， （19）， we obtain the following equations （c′） and （d′） respectively.
[1] Zhang Z L， Zhao L K. Toeplitz algebra on the Harmonic Dirichlet space. J Fudan Univ Nat Sci， 2008， 47（2）： 251-259.
[2] Lee Y J. Finite rank sums of products of Toeplitz and Hankel operators. J Math Anal Appl， 2013， 397： 503-514.
[3] Lee Y J， Zhu K H. Sums of products of Toeplitz and Hankel operators on the Dirichlet space. Integr Equ Oper Theory， 2011， 71： 275-302.
Key wordsHarmonic Dirichlet space； Small Hankel operator； Commutativity； Zero product CLC numberO 17
7.1Document codeA
Let D be the open unit disk in the complex plane C and dA be the normalized area measure on D. The Sobolev space S is the completion of the space of ooth function f on D such thatE
Recently， products of Toeplitz operators and Hankel operators on the Dirich-
let space he been studied intensively[2？9]. As has implied[10？15]， the theory of Toeplitz operators on harmonic function space is quite different from the theory on analytic function space. For the operators on harmonic Dirichlet space， （semi-） commutativity[15]and zero-product[16]of Toeplitz operators on Dhwith symbols in M he been completely characterize. In this paper， for symbols in M， we consider the problem when two all Hankel operators is commutativity. By using the matrix representation of all Hankel operator on harmonic Dirichlet space， we give a complete characterization of such operators to be commutativity. By using a similar method， the zero-product of all Hankel operators is also characterized.
2Preliminaries
Recall that U is the operator on S defined by （Uf）（z） = f（ˉz）， f∈S， and U is an unitary on S， U？= U = U？1[1]. For simplicity， for f∈S， denote？f as Uf， i.e.，？f（z） = f（ˉz）.
Let P be the orthogonal projection from S onto D and P1be the orthogonal projection from S ontoˉD， then P1= UPU[1]. Similarly， for the orthogonal projection Q， we he the following result.
As the computation in Theorem 1， by equations above， we obtain the following equations.
Compare equations （14），（18）， （19） and equations （16）， （19）， we he the following equations （a′） and （b′） respectively. Compare equation （15），（18） and equations（17），（18）， （19）， we obtain the following equations （c′） and （d′） respectively.
[1] Zhang Z L， Zhao L K. Toeplitz algebra on the Harmonic Dirichlet space. J Fudan Univ Nat Sci， 2008， 47（2）： 251-259.
[2] Lee Y J. Finite rank sums of products of Toeplitz and Hankel operators. J Math Anal Appl， 2013， 397： 503-514.
[3] Lee Y J， Zhu K H. Sums of products of Toeplitz and Hankel operators on the Dirichlet space. Integr Equ Oper Theory， 2011， 71： 275-302.