A perturbation problem for the shift semigroup (Q2461223)

The paper treats an operator theoretical problem motivated by Tsirelson's construction of uncountably many mutually non-cocycle conjugate \(E_0\)-semigroups of type III. \(E_0\)-semigroups are semigroups of unital endomorphisms of the algebra of all bounded linear operators on a separable infinite-dimensional Hilbert space, with an appropriate continuity property. The author investigates the structure of \(C_0\)-semigroups of bounded operators \(\{T_t\}_{t\geq 0}\) acting on \(L^2(0,\infty)\) and satisfying the conditions (C1) \(T_t^*S_t=I\), \(t\geq 0\) and (C2) \(T_t-S_t\) is a Hilbert-Schmidt operator, \(t\geq 0\). Here, \(\{S_t\}_{t\geq 0}\) is the shift semigroup of \(L^2(0,\infty)\). Firstly, he explicitly computes the resolvent of the generator of a \(C_0\)-semigroup \(\{T_t\}_{t\geq 0}\) satisfying (C1) and shows that \(\{T_t\}_{t\geq 0}\) is completely characterized by a certain holomorphic function on the right-half plane which is called the half-density function for \(\{T_t\}_{t\geq 0}\). Next, a complete characterization of \(C_0\)-semigroups satisfying both conditions (C1) and (C2) is presented in terms of the associated half-density functions. After this, he constructs a class of \(C_0\)-semigroups satisfying (C1) and (C2) parameterized by a function in the space \(L^1_{\text{loc}}[0,\infty)\cap L^2((0,\infty), (1\wedge x)dx)\), and describes its relationship to Tsirelson's off white noise.