On the idempotent solutions of equations 2
Aba A, Chun Yuan Deng, Bab B · 2006
Abstract. We deduce the necessary and sufficient conditions for the operator equations ABA A = 2 and BAB B = 2 to have idempotent solutions and obtain some connections between its solutions and idempotent operators. Keywords: idempotent, matrix equation, operator equation. 1. Introduction Let H be a complex Hilbert space. Denote by B () H the Banach algebra of all bounded linear operators on H . For A ,() BBH ∈ , if A and B satisfy the relations ABA A = 2 and BAB B = 2 , (1) we say the pair of (, ) AB is the solution of (1). In [1], Vidav has investigated the selfadjoint solutions of (1) and showed that the pair of (, ) AB is selfadjoint solution of (1) iff there exists unique idempotent operator P such that APP = * and B = PP * . In [2], Rakocevic gave another proof of this result by using some properties of generalized inverses. In [3], Schmoeger generalized the Vidav's result concerning (1) by using some properties of Drazin inverses. The aim of this paper is to investigate some connections between idempotent operators and the solutions of (1).