Solutions to the operator equation AX=XAX
Junlian Xu · Journal of Baoji University of Arts and Sciences · 2010
Aim To discuss the necessary and sufficient conditions for the existence of nontrivial solutions(that is X≠0,I) to the operator equation AX=XAX are discussed among which A is a bounded linear operator on a Hilbert space H.Methods The technique of block operator matrix is used to discuss the problem.Results and Conclusion The necessary and sufficient condition for the existence of nontrivial solutions(that is X≠0,I) to the operator equation AX=XAX is the operator A has nontrivial invariant subspaces in H.In addition,the new proof is given.