Approximately reducing subspaces for unbounded linear operators

Abstract

We give sufficient conditions for generation of strongly continuous contraction semigroups of linear operators on Hilbert or Banach space. Let L be a dissipative (unbounded) linear operator in a Hilbert space H and let { P n } be an increasing sequence of self-adjoint projections converging weakly to the identity projection. We show that if there is a positive integer k such that for all n the range of P n is contained in the domain of L and mapped by L into the range of P n + k , and if the sequence { LP n − P n LP n } is dominated in norm (∥ LP n − P n LP n ∥ ⩽ a n ) by some { a n } ⊂ R + with ∑ n = 1 ∞ a n −1 = ∞, then the closure of the restriction of L to ∪ n = 1 ∞ range ( P n ) is the infinitesimal generator of a strongly continuous contraction semigroup on H . Applications to an important class of finite perturbations, properly larger than the finite Kato perturbations, are given. We also give sufficient conditions for generation of contraction semigroups when { P γ } (indexed by a directed set) is a set of bounded self-adjoint operators converging weakly to the identity and each having range contained in D( L). In the latter theorem, and in an analogous theorem for dissipative linear operators L in a Banach space, we do not assume that L interchanges at most finitely many of the approximately reducing operators P γ .