Resumen de Range, kernel orthogonality and operator equations

Mohamed Amouch

• Let A be a Banach algebra and L(A) the algebra of all bounded linear operators acting on A. For a; b 2 A, the generalized derivation ±a;b 2 L(A) and the elementary operator ¢a;b 2 L(A) are de¯ned by ±a;b(x) = ax ¡ xb and ¢a;b(x) = axb ¡ x, x 2 A.

  Let da;b = ±a;b or ¢a;b. In this note we give couples (a; b) 2 A2 such that the range and the kernel of da;b are orthogonal in the sense of Birkho®. As application of this results we give consequences for certain operator equations inspired by earlier studies of the equation ® + ®¡1 = ¯ + ¯¡1 for automorphism ®; ¯ on Von Neuman algebras.