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Solvability and different solutions of the operator equation \(XAX=BX\)

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Abstract

In this paper, we present certain necessary and sufficient conditions for the existence of a nonzero solution of the operator equation \(XAX=BX.\) We present the general form of the solution and describe all cases when this equation has infinitely many solutions as well as when all the solutions are idempotent. We generalize the existing particular results that concern this equation in the special cases when \(A=B\) or \(A*\le B.\) Furthermore, we derive two possible representations of an arbitrary solution as a \(2\times 2\) operator matrix which allows us to discuss the existence of an invertible and a positive solution, but also to open the discussion to the existence of any type of solutions using related results for \(2\times 2\) operator matrices. Also, we consider the operator equation \(XAX=AX,\) which is closely related to the “invariant subspace problem” and describe the set of all right(left) invertible and all Fredholm solutions.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Kuwait University, Kuwait City, Kuwait

    Dragana S. Cvetković-Ilić

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  1. Dragana S. Cvetković-Ilić
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Correspondence to Dragana S. Cvetković-Ilić.

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Communicated by M. S. Moslehian.

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Cvetković-Ilić, D.S. Solvability and different solutions of the operator equation \(XAX=BX\). Ann. Funct. Anal. 14, 5 (2023). https://doi.org/10.1007/s43034-022-00229-x

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