Semigroups of length increasing transformations

The semigroup of all operators T such that (Tx,Tx)?(x,x), for all elements of x of a finite-dimensional complex vector space with ( , ) a given, possibly indefinite Hermitian form on that space, is the object under study. It is shown that this semigroup is closed under the operation of taking adjoints with respect to the given form and that every semigroup element may be written as a product of a unitary and a self-adjoint operator (polar form), with the unitarity and self-adjointness defined with respect to the given form. It is further shown that the semigroup is generated by the group of isometries of the given form and the union of a finite family of semigroups, each of which is of an elementary nature. The intersection of the above semigroup with the complex symplectic group is also considered; all the above results have analogues in this case.