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On the operator equations \(ABA=A^2\) and \(BAB=B^2\). (English) Zbl 1164.47309

The author investigates properties of operators \(A\) and \(B\) on Banach or Hilbert spaces, satisfying the equations \(ABA=A^2\) and \(BAB=B^2\). It is shown that, if both Drazin indices of \(A\) and \(B\) are equal to \(1\), then \(A=PQ\) and \(B=QP\) for some idempotents \(P\) and \(Q\), if and only if \(ABA=A^2\) and \(BAB=B^2\). As a corollary, the ordinary point, approximate point, residual and continuous spectrums of \(A\) and \(B\) coincide. Also, \(\lambda-A\) is Fredholm if and only if \(\lambda-B\) is Fredholm, and in this case their Fredholm indices coincide. The result becomes very interesting in the case of Hilbert space operators. Precisely, if \(A\) and \(B\) are selfadjoint and \(ABA=A^2\), \(BAB=B^2\) hold, then: \(A\) is invertible or \(0\) is a simple pole of the resolvent of \(A\); \(A\) is non-negative and its spectrum is contained in the set \(\{0\}\cup[1,\infty)\); \(A\) is Drazin invertible and its Drazin index is at most \(1\); the Drazin inverse \(A^D\) satisfies \(0\leq A^D\leq 1\); if \(A\) is nonzero, then \(\| A\| \geq 1\); if \(\| A\| =1\), then \(A^2=A=B\).As a corollary, the author proves a well-known result of I.Vidav [Publ.Inst.Math., Nouv.Sér.4(18), 157–163 (1964; Zbl 0125.35101)], which was also proved by V.Rakočević [Publ.Inst.Math., Nouv.Sér.68(82), 105–107 (2000; Zbl 1164.47315)]: \(A\) and \(B\) are selfadjoint and they satisfy \(ABA=A^2\) and \(BAB=B^2\) if and only if there exists the unique idempotent \(P\) satisfying \(A=PP^*\) and \(B=P^*P\). Finally, the author investigates the reflection of previous results to special classes of operators on Hilbert spaces, such as paranormal and hyponormal operators. The author also discusses the case of the Hermitian elements in Banach algebras.

MSC:

47A50 Equations and inequalities involving linear operators, with vector unknowns
47A53 (Semi-) Fredholm operators; index theories
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B20 Subnormal operators, hyponormal operators, etc.
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