Positive Contraction Semigroup Research Articles

H∞-calculus for semigroup generators on BMO

We prove that the negative generator L of a semigroup of positive contractions on L∞ has bounded H∞(Sη)-calculus on the associated Poisson semigroup-BMO space for any angle η>π/2, provided L satisfies Bakry-Émery's Γ2≥0 criterion. Our arguments only rely on the properties of the underlying semigroup and work well in the noncommutative setting. A key ingredient of our argument is a type of quasi monotone properties for the subordinated semigroup Tt,α=e−tLα,0<α<1, that is proved in the first part of this article.

Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one

This paper provides various “contractivity” results for linear operators of the form where C are positive contractions on real ordered Banach spaces X. If A generates a positive contraction semigroup in Lebesgue spaces , we show (M. Pierre's result) that is a “contraction on the positive cone”, i.e. for all provided that . We show also that this result is not true for 1 ⩽ . We give an extension of M. Pierre's result to general ordered Banach spaces X under a suitable uniform monotony assumption on the duality map on the positive cone . We deduce from this result that, in such spaces, is a contraction on for any positive projection C with norm 1. We give also a direct proof (by E. Ricard) of this last result if additionally the norm is smooth on the positive cone. For any positive contraction C on base‐norm spaces X (e.g. in real spaces or in preduals of hermitian part of von Neumann algebras), we show that for all where N is the canonical half‐norm in X. For any positive contraction C on order‐unit spaces X (e.g. on the hermitian part of a algebra), we show that is a contraction on . Applications to relative operator bounds, ergodic projections and conditional expectations are given.

Diffusion with nonlocal boundary conditions

We consider second order differential operators Aμ on a bounded, Dirichlet regular set Ω⊂Rd, subject to the nonlocal boundary conditionsu(z)=∫Ωu(x)μ(z,dx)for z∈∂Ω. Here the function μ:∂Ω→M+(Ω) is σ(M(Ω),Cb(Ω))-continuous with 0≤μ(z,Ω)≤1 for all z∈∂Ω. Under suitable assumptions on the coefficients in Aμ, we prove that Aμ generates a holomorphic positive contraction semigroup Tμ on L∞(Ω). The semigroup Tμ is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in C(Ω‾). We also prove that Tμ is immediately compact and study the asymptotic behavior of Tμ(t) as t→∞.

Dirichlet regularity and degenerate diffusion

Let Ω C ℝ N be an open and bounded set and let m: Ω→ (0, oo) be measurable and locally bounded. We study a natural realization of the operator mΔ in Co(Ω) := {u ∈ C(Ω): u |∂Ω = 0}. If Ω is Dirichlet regular, then the operator generates a positive contraction semigroup on C 0 (Ω) whenever 1/m ∈ L p loc (Ω) for some p > N/2. If m(x) does not go fast enough to 0 as x → ∂Ω, then Dirichlet regularity is necessary. However, if |m(x)| ≤ c · dist(x, ∂Ω) 2 , then we show that mΔ 0 generates a semigroup on C 0 (Ω) without any regularity assumptions on Ω. We show that the condition for degeneration of m near the boundary is optimal.

Cores for parabolic operators with unbounded coefficients

Let A = ∑ i , j = 1 N q i j ( s , x ) D i j + ∑ i = 1 N b i ( s , x ) D i be a family of elliptic differential operators with unbounded coefficients defined in R N + 1 . In [M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., in press], under suitable assumptions, it has been proved that the operator G : = A − D s generates a semigroup of positive contractions ( T p ( t ) ) in L p ( R N + 1 , ν ) for every 1 ⩽ p < + ∞ , where ν is an infinitesimally invariant measure of ( T p ( t ) ) . Here, under some additional conditions on the growth of the coefficients of A , which cover also some growths with an exponential rate at ∞, we provide two different cores for the infinitesimal generator G p of ( T p ( t ) ) in L p ( R N + 1 , ν ) for p ∈ [ 1 , + ∞ ) , and we also give a partial characterization of D ( G p ) . Finally, we extend the results so far obtained to the case when the coefficients of the operator A are T-periodic with respect to the variable s for some T > 0 .

ON PERTURBED POSITIVE SEMIGROUPS ON THE BANACH SPACE OF TRACE CLASS OPERATORS

Let [Formula: see text] be the generator of a positive (i.e. leaving invariant [Formula: see text]) contraction semigroup on [Formula: see text], the space of self-adjoint trace class operators on a Hilbert space H, endowed with the trace norm, and let [Formula: see text] be a positive linear operator such that [Formula: see text], [Formula: see text]. We show that there exists a minimal positive contraction semigroup generated by some [Formula: see text] and provide a systematic study of the total mass carried by individual trajectories [Formula: see text] with non-negative initial data ρ. The analysis relies on mathematical properties of two (a priori) different extensions [Formula: see text] of the functional [Formula: see text] to [Formula: see text].

Second-order operators with degenerate coefficients

We consider properties of second-order operators H = − ∑ i , j = 1 d ∂ i c i j ∂ j on ℝd with bounded real symmetric measurable coefficients. We assume that C = (cij) ⩾ 0 almost everywhere, but allow for the possibility that C is degenerate. We associate with H a canonical self-adjoint viscosity operator H0 and examine properties of the viscosity semigroup S(0) generated by H0. The semigroup extends to a positive contraction semigroup on the Lp-spaces with p ∈ [1, ∞]. We establish that it conserves probability and satisfies L2 off-diagonal bounds, and that the wave equation associated with H0 has finite speed of propagation. Nevertheless, S(0) is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor Hölder continuous. In particular, one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in C2−ε(ℝd) with ε > 0.

The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions

We investigate the Laplacian Δ on a smooth bounded open set Ω ⊂ Rn with Wentzell-Robin boundary condition \(\beta u + \frac{{\partial u}}{{\partial v}} + \Delta u = 0\) on the boundary Γ. Under the assumption β ∈ C(Γ) with β ≥ 0, we prove that Δ generates a differentiable positive contraction semigroup on \(C\left( {\bar \Omega } \right)\) and study some monotonicity properties and the asymptotic behaviour.

Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

The aim of this paper is the study of a new sequence of positive linear approximation operators Mn, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators.

Functional calculi for linear operators in vector-valued $L^p$-spaces via the transference principle

Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class $\cal H \cal T$ and $1<p<\infty$. If $Y=\mathbb{C}$, it is shown by means of the transference principle due to Coifman and Weiss that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda\pm\pi/2|<\theta\}$, where $\theta>0$ and on each sector $\Sigma_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda|<\theta\}$ with $\theta<\pi/2$, respectively. Several extensions of these results to the vector-valued case $L^p(\Omega,\mu,Y)$ are presented. In particular, let $-A$ be the generator of a bounded group on a Banach spaces of class $\cal H\cal T$. Then it is shown that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta$, $\theta > 0$, and that $-A^2$ admits an $H^\infty$-calculus on each sector $\Sigma_\theta$, where $\theta > 0$. Applications of these results deal with elliptic boundary value problems on cylindrical domains and on domains with non smooth boundary.

Lototsky-Schnabl operators on compact convex sets and their associated limit semigroups

Given a metrizable compact convex setX of a locally convex Hausdorff space, a positive projectionT:C(X, ℝ)→C(X, ℝ) and a continuous function λ:X→[0, 1], it is shown that under suitable assumptions there exists a positive contraction semigroup onC(X, ℝ) that can be represented in terms of the Lototsky-Schnabl operators associated withT and λ. Several properties of this semigroup are investigated. In particular, its infinitesimal generator is determined in a core of its domain. WhenX⊂ℝ p for somep≥1, then the generator is shown to be a degenerate elliptic second order differential operator.

Weak convergence of semigroups implies strong convergence of weighted averages

For a fixed p , 1 ⩽ p &gt; ∞ p,1 \leqslant p &gt; \infty , let { T t : t &gt; 0 } \{ {T_t}:t &gt; 0\} be a strongly continuous semigroup of positive contractions on L p {L_p} of a σ \sigma -finite measure space. We show that weak convergence of { T t : t &gt; 0 } \{ {T_t}:t &gt; 0\} in L p {L_p} is equivalent with the strong convergence of the weighted averages ∫ 0 ∞ T t f μ n ( d t ) ( n → ∞ ) \int _0^\infty {{T_t}f{\mu _n}(dt)(n \to \infty )} for every f ∈ L p f \in {L_p} and every sequence ( μ n ) ({\mu _n}) of signed measures on ( 0 , ∞ ) (0,\infty ) , satisfying sup n | | μ n | | &gt; ∞ ; lim n μ n ( 0 , ∞ ) = 1 {\sup _n}||{\mu _n}|| &gt; \infty ;{\lim _n}{\mu _n}(0,\infty ) = 1 ; and for each d &gt; 0 , lim n sup c ⩾ 0 | μ n | ( c , c + d ] = 0 d &gt; 0,{\lim _n}{\sup _{c \geqslant 0}}|{\mu _n}|(c,c + d] = 0 . The positivity assumption is not needed if p = 1 p = 1 or 2. We show that such a result can be deduced-not only in L p {L_p} , but in general Banach spaces-from the corresponding discrete parameter version of the theorem.

A maximal ergodic theorem for Abel means of continuous-parameter operator semigroups

A positive contraction semigroup { T t : t ⩾ 0} of type C 1 in the L 1 space of a σ-finite complete measure space (Ω, β, μ) is studied, If J λ is the resolvent of { T t } and if ƒ ϵ L 1 then, for a suitable version of (J λƒ)(ω) , ƒ ∗ is defined by ƒ ∗(ω) = sup{(λJ λƒ)(ω): 0 < λ < ∞} . The theorem of the title then states that ∝ Aƒdμ ⩾ 0 when A = {ω: ƒ ∗(ω) > 0 . When { T t } is a sub-Markovian semigroup this result implies the almost everywhere existence and finiteness of lim λ → 0+ (λJ λƒ)(ω) .