Semigroups of Partial Isometries and Symmetric Operators

Let ({{{ V(t) | t in [0 , infty) }}}) be a one-parameter strongly continuous semigroup of contractions on a separable Hilbert space and let V(?t) : = V*(t) for ({t in [0, infty)}). It is shown that if V(t) is a partial isometry for all ({t in [-t_0 , t_0], t_0 > 0}), then the pointwise two-sided derivative of V(t) exists on a dense domain of vectors. This derivative B is necessarily a densely defined symmetric operator. This result can be viewed as a generalization of Stone’s theorem for one-parameter strongly continuous unitary groups, and is used to establish sufficient conditions for a self-adjoint operator on a Hilbert space ({mathcal{K}}) to have a symmetric restriction to a dense linear manifold of a closed subspace ({mathcal H subset mathcal K}). A large class of examples of such semigroups consisting of the compression of the unitary group generated by the operator of multiplication by the independent variable in ({mathcal {K} := oplus _{i=1} ^n L^2 (mathbb {R})}) to certain model subspaces of the Hardy space of n?compenent vector valued functions which are analytic in the upper half plane is presented.