The solution set to the nonlinear operator equation <em>XAX</em> = <em>BX</em> on Hilbert spaces

Research Article

The solution set to the nonlinear operator equation XAX = BX on Hilbert spaces

Published in: Quaestiones Mathematicae
Volume 47 , issue 5 , 2024 , pages: 949–959
DOI: 10.2989/16073606.2023.2274586
Author(s): Hua Wang Inner Mongolia University of Technology, PRC , Jingsong Wu Inner Mongolia University of Technology, PRC

Abstract

We present the representation of solution sets of the operator equation XAX = BX on Hilbert spaces by taking advantage of the invariant subspace of B, including solution sets of nonzero solutions, normal solutions and self-adjoint solutions. Moreover, necessary conditions and sufficient conditions are established for existence of nonzero solutions of the operator equation XAX = BX which commute with B.

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