Nearly invariant subspaces for shift semigroups

Let{T(t)}_(t≥0) be a C_(0)-semigroupon an infinite-dimensional separable Hilbert space;a suitable definition of near{T(t)^(*)}_(t≥0) invariance of a subspace is presented in this paper.A series of prototypical examples for minimal nearly{S(t)^(*)}_(t≥0) invariant subspaces for the shift semigroup{S(t)}_(t≥0) on L^(2)(0,∞)are demonstrated,which have close links with near T_(θ)^(*)invariance on Hardy spaces of the unit disk for an inner functionθ.Especially,the corresponding subspaces on Hardy spaces of the right half-plane and the unit disk are related to model spaces.This work further includes a discussion on the structure of the closure of certain subspaces related to model spaces in Hardy spaces.