Riccati Equation Related To Toeplitz Operator In Bergman Space

Since the Riccati equation was proposed,it has always attracted the attention of many scholars.Until now,the existence of the solution of the equation has been continuously improved.The operator Riccati equation is defined as XAX+XB-CX-D=0,Where A,B,C,D are the concrete operator in Hilbert space.This paper will study Riccati equation from two aspects.First,we discuss the Toeplitz operator on Bergman space which satisfies Riccati equation.The relationship between invariant subspace and the special form of Riccati equation XAX+AX=0 is also studied.Second,the problem of the existence of the solution of the special form of Riccati equations XB-CX=0 is transformed into the condition for the existence of complex symmetric operators.This paper describe the Toeplitz operators on Bergman Spaces and weighted composition differentiation operators on Hardy Spaces,Bergman Spaces and derivative Hardy Spaces satisfy the condition that they are complex symmetric operators.Furthermore,the conditions of self-adjoint operators are given.And also found the relation between the self-adjointness and normality of the weighted composition differentiation operator on the Bergman space.