Differentiability of the age-dependent population system with time delay in the birth process

In this paper, by using semigroup approach, we concerned with the regularity of the age-dependent population system with instantaneous time delay in the birth process. We prove that, under some additional condition, the solution semigroup of the population evolution equation is eventually differentiable, i.e., there exists t₀>0 such that T(t) is differentiable for t>t₀.     

TOWARDS A CHARACTERIZATION OF EVENTUALLY NORM CONTINUOUS SEMIGROUPS ON BANACH SPACES

Abstract

We characterize the condition

(E) lim_μ→±∞ ∥R(iμ,A∥ = 0

for a generator A of an exponentially stable semigroup (T(t))_t≥0 on an arbitrary Banach space in terms of (T(t))_t≥0. As shown earlier on Hilbert spaces this condition is equivalent to the norm continuity of the semigroup for t > 0.     

On supercyclicity of operators from a supercyclic semigroup

We show that for every supercyclic strongly continuous operator semigroup {Tt}_t?0 acting on a complex F-space, every Tt with t>0 is supercyclic. Moreover, the set of supercyclic vectors of each Tt with t>0 is exactly the set of supercyclic vectors of the entire semigroup.     

On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space

We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C ₀ semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.      

On the stability of differentiability of semigroups

LetA be the infinitesimal generator of aC ₀-semigroup. The semigroup generated byA is called differentiable ifA exp (At) is bounded for everyt>0. In this note, an example is given of an operatorA and a bounded operatorB such that the semigroup generated byA is differentiable but the semigroup generated byA+B is not. This gives a negative answer to a question of Pazy.     

Some dichotomy results for central Gaussian convolution semigroups on compact groups

 Consider a central Gaussian convolution semigroup (μ _t ) _{t > 0} on a connected compact semisimple group. Then either the measure μ _t is singular with respect to Haar measure for all t > 0, or there exists a time t such that μ _t is absolutely continuous with respect to Haar measure and admits a continuous density. Received: 6 November 2001 / Revised version: 13 June 2002 / Published online: 30 September 2002 Research partially supported by NSF grant DMS 0102126 Mathematics Subject Classification (2000): 28C10, 28C20, 60B15, 60G30 Keywords or phrases: Gaussian convolution semigroups – Dichotomy – Absolute continuity     

On the behavior of orbits of uniformly stable semigroups at infinity

For uniformly stable bounded analytic C ₀-semigroups {T(t)} _t≥0 of linear operators in a Banach space B, we study the behavior of their orbits T (t)x, x ∈ B, at infinity. We also analyze the relationship between the order of approaching the orbit T (t)x to zero as t → ∞ and the degree of smoothness of the vector x with respect to the operator A ⁻¹ inverse to the generator A of the semigroup {T(t)} _t≥0. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than , where a > 0, 0 < α < π/(2(π-θ)), θ is the angle of analyticity of {T(t)} _t≥0, and the collection of these orbits is dense in the set of all orbits. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 148–159, February, 2006.     

Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

On the spectral mapping theorem for perturbed strongly continuous semigroups

We consider a strongly continuous semigroup (T(t))_{t \geqq 0}(T(t))_{t \geqq 0} with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup (S(t))_{t \geqq 0}(S(t))_{t \geqq 0} generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t))_{t \geqq 0}(S(t))_{t \geqq 0} .     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

On the structure of a semigroup of operators with finite-dimensional ranges

In the present paper, we describe the structure of a strongly continuous operator semigroup T(t) (where T: ?₊ → End X and X is a complex Banach space) for which ImT(t) is a finite-dimensional space for all t > 0. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a C ₀-semigroup.     

Smoothing properties of nonlinear transition semigroups: case of lipschitz nonlinearities

We consider a semilinear stochastic differential equation in a Hilbert space H with a Lipschitz continuous (possibly unbounded) nonlinearity F. We prove that the associated transition semigroup {P_t, t ≥ 0}, acting on the space of bounded measurable functions from H to , transforms bounded nondifferentiable functions into Fréchet differentiable ones. Moreover we consider the associated Kolmogorov equation and we prove that it possesses a unique “strong” solution (where “strong” means limit of classical solutions) given by the semigroup {P_t, t ≥ 0} applied to the initial condition. This result is a starting point to prove existence and uniqueness of strong solutions to Hamilton - Jacobi - Bellman equations arising in control theory. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Extremal properties of contraction semigroups onc 0

For any complex Banach spaceX, letJ denote the duality mapping ofX. For any unit vectorx inX and any (C ₀) contraction semigroup (T _t ) _t>0 onX, Baillon and Guerre-Delabriere proved that ifX is a smooth reflexive Banach space and if there isx ^*∈J(x) such that ÷〈(T(t)x, J(x)〈÷→1 ast→∞, then there is a unit vectory∈X which is an eigenvector of the generatorA of (T _t ) _t>0 associated with a purely imaginary eigenvalue. They asked whether this result is still true ifX is replaced byc ₀. In this article, we show the answer is negative Partial results of this paper were obtained when the author attended the International Conference of Convexity at the University of Marne-La-Vallée. He would like to express his gratitude for the kind hospitality offered to him. He would also like to thank Profs. Goldstein and Jamison for their valuable suggestions.     

Two examples of local ergodic divergence

This note contains the first example of a 1-parameter semigroup {T _pt≧0} of linear contractions inL _p (1<p<∞) for which the assertion of the local ergodic theorem (t ¹∝₀′T _sfds conv. a.e. ast → 0+0 for allf ∈L _p) fails to be true. The first example is a continuous semigroup of unitary operators inL ₂, the second a power-bounded continuous semigroup of positive operators inL ₁. This answers problems of Kubokawa, Fong and Sucheston. In memory of our friend and colleague Shlomo Horowitz     

On exponential stability of contraction semigroups

A semigroup [T(t)] on a Hilbert space is exponentially stable if there exist real constants M≥1 and α>0 such that ∥T(t)∥≤Me ^−αt for every t≥0. If [T(t)] is a strongly continuous contraction semigroup, then it is proved that we can set M=1 in the definition of exponential stability if and only if the generator A of [T(t)] is boundedly strict dissipative (just a strict dissipative A is not enough).     

Approximating a Feller Semigroup by Using the Yosida Approximation of the Symbol of its Generator

We show that if the pseudodifferential operator −q(x,D) generates a Feller semigroup (T_t)_t≥0 then the Feller semigroups (T_t^(v))_t≥0 generated by the pseudodifferential operators with symbol will converge strongly to (T_t)_t≥0 as ν →∞.     

On bounded solutions of abstract differential equations

Summary We construct nontrivial bounded solutions of an abstract evolution equationu'(t)=Au(t) (—∞<t<∞) whereA generates a (C ₀) semigroup of operators {T _t;t≥0} such thatT _t converges strongly to zero ast→∞.

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Riassunto Si costruiscono soluzioni limitate non triviali di una equazioneu'(t)=Au(t) (—∞<t<∞) doveA è il generatore di un semigruppo di operatori {T _t;t≥0} tale cheT _t converge fortemente verso 0 pert→∞.

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Supported by National Science Foundation grant GP-12722.     

Subexponential ultracontractivity and L p ? L q functional calculus

 We prove that if a symmetric submarkovian semigroup (T _t ) _t>0 satisfies an estimate of the form

Atoms and regularity for measures in a partially defined free convolution semigroup

Consider a Borel probability measure on the real line, and denote by {_t : t1} the free additive convolution semigroup defined by Nica and Speicher. We show that the singular part of _t is purely atomic and the density of _t is locally analytic, provided that t > 1. The main ingredient is a global inversion theorem for analytic functions on a half plane.Mathematics Subject Classification (2000): 46L54, 30A99Supported in part by a grant from the National Science Foundation.