On The Properties Of Canonical Solution Operator To (?) Restricted To Dirichlet Space

The canonical solution operators is a class of important operator related to Neumann equation.And a series of properties of this class operator on Bergman spaces has been researched by many scholars.This paper mainly studies some properties of the canonical solution operator to(?)restricted to Dirichlet space.The main results and innovation points are as follows:1.Introduced the definition of the Dirichlet space and some properties of the canon-ical solution operator to(?);2.Given the integral representation of the canonical solution operator to(?)restricted to Dirichlet space.For any fix(p,q)-form ?,obtained the representation of(p,q-1)-form u such that(?)u=?;3.Discussed the properties of the canonical solution operator to(?)restricted to Dirichlet space of the unit disk,and proved that the operator S1 is a Hilbert-Schmidt operator;4.Proved that the operator Sk(k? 2,n? k)is a Hilbert-Schmidt operator on Dirichlet space of the unit disk;5.Proved that the canonical solution operator S1 and the operator Sk(k?2,n?k)are not Hilbert-Schmidt operator on Dirichlet space of the bidisk;6.Proved that the concomitant operator Pn(n?2)is similar to(?)P1 on Dirichlet space.