The function "f"("z")=log "z" can be defined on any simply connected open set in the complex plane not containing the origin, and it is holomorphic on such a domain. This implies that one can define ln "T" as long as the spectrum of "T" does not contain the origin and there is a path going from the origin to infinity not crossing the spectrum of "T" (e.g., if the spectrum of "T" is a circle with the origin inside of it, it is impossible to define ln "T").