  The Roper--Suffridge extension operator and its modifications are powerful
tools to construct biholomorphic mappings with special geometric properties.
  The first purpose of this paper is to analyze common properties of different
extension operators and to define an extension operator for biholomorphic
mappings on the open unit ball of an arbitrary complex Banach space. The second
purpose is to study extension operators for starlike, spirallike and convex in
one direction mappings. In particular, we show that the extension of each
spirallike mapping is $A$-spirallike for a variety of linear operators $A$.
  Our approach is based on a connection of special classes of biholomorphic
mappings defined on the open unit ball of a complex Banach space with
semigroups acting on this ball.
