On the abstract theorem of Picard

Let $A$ be a complex Banach algebra with unit. It was shown by Williams [1] that elements $\mathbf a,\mathbf b\in A$ commute if and only if $\sup\limits_{\lambda\in\mathbf C}\|\exp(\lambda\mathbf b)\mathbf a\exp(-\lambda\mathbf b)\|<\infty$. 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 [1] and Khasbardar et Thakare [2] are refined by using [3, 4, 5]. An abstract version of Picard theorem is obtained in this context.