Some properties of operator exponent

We study operators given by series, in particular, operators of the form $e^B=\sum\limits_{n=0}^{\infty}{B^n}/{n!},$ where $B$ is an operator acting in a Banach space $X$. A corresponding example is provided. In our future research, we will use these operators for introducing and studying functions of operators constructed (with the use of the Cauchy integral formula) on the basis of scalar functions and admitting a faster than power growth at infinity.