We assume that there exists a semigroup of positive operators on Lp satisfying a monotone property but do not assume any geometric/metric structure on Ω.
Throughout the rest of the paper, we suppose that the semigroup is hyperbolic; that is, there exist a projection and constants such that commutes with, is invariant with respect to, is invertible, and the following hold: (4.5).
Does there exist a limit?
Let C ( J, X ), be the Banach space of continuous functions from J into X with the usual supremum norm ∥ x ∥ C : = sup t ∈ J ∥ x ( t ) ∥, for x ∈ C. In this paper, we assume that A : D ( A ) ⊂ X → X is the infinitesimal generator of a strongly continuous semigroup T , then there exists a constant M ≤ 1.
The conservativity of a minimal quantum dynamical semigroup is proved whenever there exists a "generalized" subharmonic operator bounded from below by the dissipative part of the infinitesimal generator.
(5.1) Then, from [25], A generates a compact analysis semigroup (S t)) ((tgeq 0)) in X and there exists a constant (Mgeq1) such that (|S t)| leq M).
We prove that ifSis an amenable semigroup and S={Tt: t∈S} is a nonexpansive semigroup on a closed, convex subsetCin a uniformly convex Banach spaceEsuch that the setF(S) of common fixed points of S is nonempty, then there exists a nonexpansive retractionPfromContoF(S) such thatPTt=TtP=Pfor eacht∈SandPx∈co{Ttx: t∈S} for eachx∈C.
Intrinsic ultracontractivity is proved for the semigroup of H when D is Hölder domain of order 0 or a uniformly Hölder domain of order α for 0 < α < 2. For every α ⩾ 2, there exists a uniformly Hölder domain of order α for which the Dirichlet Laplacian is not intrinsically ultracontractive.
If the Julia set J (G) contains an isolated point (say a), then there exists a neighbourhood (Omega _a) of a such that (Omega _a {a} subset F G)) and (psi in G) which satisfies (Omega _a subset subset psi (Omega _a).) In particular, if G is a semigroup generated by proper maps, then (psi ^{-1}(a)=a).
Total asymptotically strictly pseudocontractive semigroup if there exist a bounded function λ : [ 0, ∞ ) → ( 0, ∞ ) and sequences { μ n } ⊂ [ 0, ∞ ) and { ξ n } ⊂ [ 0, ∞ ) with μ n → 0 and ξ n → 0 as n → ∞.
