Range, kernel orthogonality and operator equations

Abstract

Let A be a Banach algebra and ℒ(A) the algebra of all bounded linear operators acting on A. For a; b ϵ A, the generalized derivation δ ₐ;ь ϵℒ(A) and the Elementary operator Δₐ;ь ϵ ℒ(A) are defined by δₐ;ь(𝓍) = a𝓍 - 𝓍b and Δₐ;b(𝓍) = a𝓍b - 𝓍, 𝓍 ϵ ₐ;ь = δ ₐ;ь or Δ ₐ;ь. In this note we give couples (ₐ;ь) ϵ A² such that the range and the dₐ;ь kernel of ₐ;ь are orthogona in the sense of Birkhoff. As application of this results we give consequences for certain operator equations inspired by earlier studies of the equation 𝛼+α⁻¹=β+ β for automorphism 𝛼; β on Von Neuman algebrasβ for automorphism 𝛼; β on Von Neuman algebras.