调和Dirichlet空间上小Hankel算子的乘积(英文)

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AbstractIn this paper， we study small hankel operators with harmonic symbols on the harmonic dirichlet space， completely characterize the condition for the cummutativity and zero-product of small Hankel operators.

Key wordsHarmonic Dirichlet space； Small Hankel operator； Commutativity； Zero product CLC numberO 177.1Document 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 smooth function f on D such that

E

Recently， products of Toeplitz operators and Hankel operators on the Dirich-

let space have 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 have been completely characterize. In this paper， for symbols in M， we consider the problem when two small Hankel operators is commutativity. By using the matrix representation of small 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 small 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 have 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 have 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.

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