Ergodic theorems for contraction semi-groups

We examine positive semigroups acting on Banach lattices and operator algebras. In the lattice framework we characterize strict positivity and strict ordering of holomorphic semigroups by irreducibility criteria. In the algebraic setting we derive ergodic criteria for irreducibility and discuss various aspects of strict positivity. Finally we examine invariant states of a C*-dynamical system in which the automorphism group is replaced by a strongly positive semigroup. We demonstrate that ergodic states are characterized by a cluster property despite the absence of a covariant implementation law for the semigroup.

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“…Kubokawa (1975) has demonstrated that if S is a C 0 -semigroup acting on L\x; d\i) then there exists a unique positive C 0 -semigroup S such that…”

Section: That If S > T Then Every S-invariant Banach Ideal Is T-invarmentioning

confidence: 99%

Strictly positive and strongly positive semigroups

Majewski

Robinson

1983

J Aust Math Soc A

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“…However, it was proven in [16] that every quasicontraction semigroup on an L 1 space has a minimal dominating positive semigroup, called the modulus semigroup, which itself is quasicontractive. Hence, in principle, one can prove stability results even in the case of a non-positive governing semigroup, by perturbing the semigroup generator with a positive operator such that the perturbed generator does indeed generate a positive semigroup.…”

Section: Discussionmentioning

confidence: 99%

“…corresponds to the partial differential equation 16) subject to the boundary condition (3.2). We solve easily equation (3.16) using the method of characteristics.…”

Section: Remark 10 We Note That the Operator A + B + C + D Is In Genmentioning

confidence: 99%

Semigroup Analysis of Structured Parasite Populations

Farkas

Green

Hinow

2010

Math. Model. Nat. Phenom.

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Abstract. Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.

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“…Let {P(i)\ t > 0} be a strongly continuous submarkovian semigroup which dominates {T(t): t > 0} ( [4], [6]). If / G L,+(/i) then P: f ^ e-'P(t)f(x) defines a linear contraction mapping from L,(/x) to LX(R+ X X, dp), where dp = dt X dp..…”

Section: Means That (I) T(t + S) = T(t)t(s) S T > 0; (Ii) ||T(0||i mentioning

confidence: 99%

“…Additionally, there is a jn-null set E(f), independent of a > 0, outside which Jq T(t)f(x) dt exists and, as a function of x, is in the equivalence class of /S T(t)fdt for every a > 0. We define , [5], [6], [8], [9]). T. Terrell [10] extended the local ergodic theorem for one-parameter submarkovian semigroups to the n-parameter case.…”

mentioning

confidence: 99%