Semigroups of partial isometries and symmetric operators
From the author's abstract: Let \(\{ V(t)\mid t \in [0 , \infty) \}\) 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 selfadjoint 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 \(n\)-component vector-valued functions which are analytic in the upper half plane is presented.
strongly continuous one parameter operator semigroups
scientific article; zbMATH DE number 5909278