Solutions to a Class of Nonlinear Operator Equations
The problems related to the solution of operator equation X2AX+XAX=BX in Hilbert space are discussed.Firstly,the sufficient conditions for the existence of rank-one solutions and rank-one idempotent commutative solutions of the operator equations at B=2A are given by using the eigenvalues of the operators.Furthermore,the necessary and sufficient conditions for the existence of finite rank solutions and commutative finite rank solutions of operator equations are given when operators have eigenmatrices.Sec-ondly,the necessary and sufficient conditions for the solvability of the operator equation are given under certain conditions.
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