Ergodic semigroups of positivity preserving self-adjoint operators

We prove that an ergodic semigroup of positivity preserving self-adjoint operators is positivity improving. We also present a new proof (using Markov techniques) of the ergodicity of semigroups generated by spatially cutoff P(?)₂ Hamiltonians.     

On perturbations preserving the immediate norm continuity of semigroups

We show that the Desch-Schappacher perturbation and the Miyadera-Voigt perturbation of an immediately norm continuous semigroup are immediately norm continuous. We also show that a perturbation theorem of C. Batty, C. Kaiser and L. Weis based on a generation theorem of A.M. Gomilko, D.-X. Feng and D.-H. Shi also preserves the immediate norm continuity of semigroups. The novelty of these results is that, contrary to the numerous related results, we obtain the immediate norm continuity of the perturbed semigroup without additional assumptions.     

Eventually norm continuous semigroups on Hilbert space and perturbations

The eventually norm continuous semigroups on Hilbert space and perturbation are studied in this paper. By resolvent of infinitesimal generator, the sufficient and necessary conditions for eventually norm continuous semigroups are given. Using the result obtained, it is proved that if is infinitesimal generator of an eventually norm continuous semigroup T(t), then there is a subspace Ξ_A of such that, for any , the semigroup S(t) generated by preserves the property of T(t).     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

Nonlinear perturbations of positive semigroups

Communicated by J. A. Goldstein     

On the spectrum of Markov semigroups via sample path large deviations

The essential spectral radius of a sub-Markovian process is defined as the infimum of the spectral radiuses of all local perturbations of the process. When the family of rescaled processes satisfies sample path large deviation principle, the spectral radius and the essential spectral radius are expressed in terms of the rate function. The paper is motivated by applications to reflected diffusions and jump Markov processes describing stochastic networks for which the sample path large deviation principle has been established and the rate function has been identified while essential spectral radius has not been calculated.     

Singular perturbations of elliptic problems

Summary The paper treates applications of singular perturbations of variational inequalities, to differential problems. Some informations on the boundary layer phenomenon are obtained. Entrata in Redazione il 20 ottobre 1971.     

Form-bounded perturbations of generators of sub-Markovian semigroups

We develop perturbation theory of generators of sub-markovian semigroups by relatively form-bounded perturbations. The L ^p-smoothing properties of semigroups and the uniqueness problem are considered. Applications to operators of mathematical physics are given.     

Norm-ideal perturbations of one-parameter semigroups and applications

The notion of equivalence classes of generators of one-parameter semigroups based on the convergence of the Dyson expansion can be traced back to the seminal work of Hille and Phillips, who in Chapter XIII of the 1957 edition of their Functional Analysis monograph, developed the theory in minute detail. Following their approach of regarding the Dyson expansion as a central object, in the first part of this paper we examine a general framework for perturbation of generators relative to the Schatten-von Neumann ideals on Hilbert spaces. This allows us to develop a graded family of equivalence relations on generators, which refine the classical notion and provide stronger-than-expected properties of convergence for the tail of the perturbation series. We then show how this framework realises in the context of non-self-adjoint Schrödinger operators.     

Kernel estimates for perturbations of positive semigroups

Given a positive C⁰-semigroup T⁰ on L₂(Ω, m) with a kernel k⁰, where (Ω, m) is a σ-finite measure space, we study a suitably perturbed semigroup T and prove existence of a kernel k for T and an estimate of the k in terms of k⁰. In this way we extend a heat kernel estimate proven by Barlow, Grigor’yan and Kumagai [4] for Dirichlet forms perturbed by jump processes. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive semigroups and perturbations of boundary conditions

Positivity - We present a generation theorem for positive semigroups on an $$L^1$$ space. It provides sufficient conditions for the existence of positive and integrable solutions of...     

Singular perturbations of abstract wave equations

Given, on the Hilbert space H₀, the self-adjoint operator B and the skew-adjoint operators C₁ and C₂, we consider, on the Hilbert space H?D(B)⊕H₀, the skew-adjoint operator

     

Singular perturbations of some nonlinear problems

We deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as ε → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles.