Solutions to the Operator Equation (A~*)~nX+X~*A~n=B
Deng Xue-hui · Journal of Southwest University · 2010
Let A and B be two bounded linear operators on a Hilbert space H. When the range space of A is closed,the operator equation (A*)nX+X*An=B is investigated by using the block operator matrix technique,the sufficient and necessary conditions for the existence of solutions of the operator equation and the representation of the solutions are established; particularly,when n=1,the operator equation A*X+X*A=B is investigated,the sufficient and necessary conditions for the existence of the positive solutions of the operator equation and the representation of the positive solutions are established.