Journal article
Approximately reducing subspaces for unbounded linear operators
Journal of functional analysis, Vol.23(4), pp.392-414
1976
DOI: 10.1016/0022-1236(76)90065-3
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
.
Details
- Title: Subtitle
- Approximately reducing subspaces for unbounded linear operators
- Creators
- Palle E.T Jørgensen - University of Pennsylvania
- Resource Type
- Journal article
- Publication Details
- Journal of functional analysis, Vol.23(4), pp.392-414
- DOI
- 10.1016/0022-1236(76)90065-3
- ISSN
- 0022-1236
- eISSN
- 1096-0783
- Publisher
- Elsevier Inc
- Language
- English
- Date published
- 1976
- Academic Unit
- Mathematics
- Record Identifier
- 9984240763102771
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