On the abstract theorem of Picard
(Lobachevskii Journal of Mathematics, Volume XVIII)
Let A be a complex Banach algebra with unit. It was shown by Williams [W] that elements a, b ∈A commute if and only if supλ ∈ C | exp (λ b) a exp (-λ b) | <∞. This result allows us to obtain an analog of the von Neumann-Fuglede-Putnam theorem in case of normal elements in a complex Banach algebra. In the present paper the results by Williams [W] and Khasbardar et Thakare [Kh] are refined by using [G,Ka,Ka2]. An abstract version of Picard theorem is obtained in this context.