  A unitary representation of a, possibly infinite dimensional, Lie group $G$
is called semibounded if the corresponding operators $i\dd\pi(x)$ from the
derived representation are uniformly bounded from above on some non-empty open
subset of the Lie algebra $\g$ of $G$. A hermitian Lie group is a central
extension of the identity component of the automorphism group of a hermitian
Hilbert symmetric space. In the present paper we classify the irreducible
semibounded unitary representations of hermitian Lie groups corresponding to
infinite dimensional irreducible symmetric spaces. These groups come in three
essentially different types: those corresponding to negatively curved spaces
(the symmetric Hilbert domains), the unitary groups acting on the duals of
Hilbert domains, such as the restricted Gra\ss{}mannian, and the motion groups
of flat spaces.
