A characterization of strongly continuous groups of linear operators on a Hilbert space

It is proved that the infinitesimal generator A of a strongly continuous se migroup of Linear operators on a Hilbert space also generates a strongly co ntinuous group if and only if the resolvent of -A, (lambda + A)(-1), is als o a bounded function on some right-hand-side half plane of complex numbers, and converges strongly to zero as the real part of lambda tends to infinit y. An application to a partial differential equation is given.