Operator matrix representations for solutions of Douglas equations
Yan-Ni Dou · Xi'an Gongcheng Keji Xueyuan xuebao · 2009
The geometrical structure of Douglas equations are studied.By using the technique of block operator,the operator matrix representations of the reduced solutions and the hermitian solutions of the operator equation BX=C are obtained and the alternative proofs of some results given by Arias,Corach and Gonzalez are also given.The results show that,under the corresponding space decompositions,the reduced solution XM with respect to the subspace M and the hermitian solution X of the operator equation BX=C have the operator matrix forms XM=(B-1MC11000),X=(B-11C1B-11C2(B-11C2)*X4),respectively.Moreover,a necessary and sufficient condition for the existence of the positive solution of the equation is that BB′C=C,BC*∈B(K)is self-adjoint,B-11C1 is positive,and R(B-11C2)R((B-11C1)1/2).