Semigroups of Partial Isometries and Symmetric Operators

Abstract

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.