Suppose ${G₁\left(t\right)}_{t\ge 0}$ and ${G₂\left(t\right)}_{t\ge 0}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators for...

We define the transient and recurrent parts of a quantum Markov semigroup 𝓣 on a von Neumann algebra 𝓐 and we show that, when 𝓐 is σ-finite, we can write 𝓣 as the sum of such semigroups. Moreover, if 𝓣 is the countable direct sum of irreducible semigroups each with a unique faithful normal invariant state ρₙ, we find conditions under which any normal invariant state is a convex combination of ρₙ's.

The Yosida approximation is treated as an inversion formula for the Laplace transform.

Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on ℬ(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on ℬ(h) of the form $⟨x,y{⟩}_s:=tr({\rho }^{1-s}x*{\rho }^{s}y)$ (s ∈ [0,1]) in order...

We show that a stochastic operator acting on the Banach lattice $L^1\left(m\right)$ of all $m$-integrable functions on $(X,𝒜)$ is quasi-compact if and only if it is uniformly smoothing (see the definition below).

Given a family of (W) contractions $T_1,...,T_N$ on a reflexive Banach space X we discuss unrestricted sequences $T_{r_n}\circ ...\circ T_{r_1}\left(x\right)$. We show that they converge weakly to a common fixed point, which depends only on x and not on the order of the operators $T_{r_n}$ if and only if the weak operator closed semigroups generated by $T_1,...,T_N$ are right amenable.

This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u^{\text{'}}\left(t\right)=Au\left(t\right)+\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}x$, t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x in ${(\lambda -A)}^{-1}C\left(X\right)$.

The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra ${}_C^{\text{'}}(ℝⁿ;M_{m\times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G{\in }_C^{\text{'}}(ℝⁿ;M_{m\times m})$ is the generating distribution of an i.d.c.s. if and only if the operator ${\partial }_t{\otimes }_{m\times m}-G\ast$ on ${ℝ}^{1+n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

Soit $X$ un espace localement compact. Tout opérateur dissipatif de domaine dense dans $C^0\left(\right(X)$ est limite d’opérateurs dissipatifs bornés. Ce résultat permet, dans le cas où $X$ est un espace homogène, de démontrer que tout opérateur dissipatif, de domaine dense et invariant sur $C^0\left(X\right)$ se prolonge en le générateur infinitésimal d’un semi-groupe à contraction invariant sur $C^0\left(X\right)$.À tout opérateur $A$ vérifiant le principe du maximum positif sur $C^0(X,\mathbf{R})$ et de domaine assez riche, on associe un opérateur bilinéaire $B$, appelé...