Regularity properties of semigroups generated by some Fleming-Viot type operators

We study different qualitative properties of the semigroup generated by some degenerate differential elliptic operators on the standard simplex of Rd. Some methods are new and are based on the representation formulas of the semigroup in terms of iterates of suitable positive operators. The main result is the ultracontractivity property which is obtained in the setting of weighted Lp-spaces. We describe the asymptotic behavior of the semigroup and obtain the compactness property in the same setting and also in spaces of continuous functions.     

Purely imaginary fractional powers of operators

Hydromechanics Institute, Academy of Sciences of the Ukrainian SSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 78–80, April–June, 1991.     

Polynomials generated by linear operators

We study the class of Banach algebra-valued -homogeneous polynomials generated by the powers of linear operators. We compare it with the finite type polynomials. We introduce a topology on similar to the weak topology, to clarify the features of these polynomials.

     

Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

Semigroup Forum - We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$mathrm {C}(partial M)$$ of continuous functions on the boundary...     

Decidability of finite quasivarieties generated by certain transformation semigroups

No Abstract. Received January 2, 2000; accepted in final form August 28, 2000.     

A note on fractional derivatives and fractional powers of operators

Definitions of fractional derivatives and fractional powers of positive operators are considered. The connection of fractional derivatives with fractional powers of positive operators is presented. The formula for fractional difference derivative is obtained.     

On semigroups generated by differential operators on Lie groups

Let dR be the differential of a strongly continuous representation of a Lie group G on a Hilbert space H. Let P be the left-invariant second-order differential operator on G with positive semidefinite main part P₂ and with first-order part P₁. If Im(P₁) is in some sense subordinate to P₂ then dR(P) is a pregenerator of a strongly continuous semigroup of operators in H. If the whole P₁ is in some sense subordinate to P₂ then that semigroup is holomorphic or, even more, dR(P) is a pregenerator of a cosine operator function.     

On uniqueness of fractional powers of multi-valued linear operators and the incomplete Cauchy problem

We obtain uniqueness of additive families {A _t } _t>0 of fractional powers of a multi-valued sectorial linear operator A ?? A ₁ in a Banach space, satisfying a certain kind of continuity with respect to the exponent and a spectral property, from uniqueness of the solutions of the second-order incomplete Cauchy problem associated with A. We show the close relationship between the multiplicativity and the uniqueness of fractional powers.     

Continuity properties of fractional powers,of the logarithm,and of holomorphic semigroups

We establish the continuity of the functions mentioned in the title (viewed as functions mapping operators onto operators) with respect to the gap and related finer topologies. Improved versions of results of Hille/Phillips and Kato concerning perturbations of holomorphic semigroups are easy consequences of our results.     

Compactness properties of certain integral operators related to fractional integration

Suppose 1≤p,q≤∞ and α > (1/p−1/q)₊. Then we investigate compactness properties of the integral operator when regarded as operator from L_p[0,1] into L_q[0,1]. We prove that its Kolmogorov numbers tend to zero faster than exp(−c_αn^1/2). This extends former results of Laptev in the case p=q=2 and of the authors for p=2 and q=∞. As application we investigate compactness properties of related integral operators as, for example, of the difference between the fractional integration operators of Riemann–Liouville and Weyl type. It is shown that both types of fractional integration operators possess the same degree of compactness. In some cases this allows to determine the strong asymptotic behavior of the Kolmogorov numbers of Riemann–Liouville operators. In memoria of Eduard (University of the West Indies) who passed away in October 2004.     

Continuous random walks and fractional powers of operators

We derive a probabilistic representation for the Fourier symbols of the generators of some stable processes. This short paper represents a bridge between probabilists and researchers working in PDE?s.     

Time-decay of semigroups generated by dissipative Schrödinger operators

We establish a representation formula for semigroups of contractions in terms of a global limiting absorption principle. As applications, we prove long-time asymptotics and dispersive estimate of the semigroup generated by −iH where H is a dissipative Schrödinger operator.     

A note on immersions of domains of fractional powers of certain sectorial operators in Sobolev spaces

In this note we give a proof of a result on immersions of domains of fractional powers of certain sectorial operators associated to strongly elliptic operators in Sobolev spaces; such immersions preserve information on fractional derivatives. We also briefly comment on the application of this result to a problem of optimal control of mosquito populations.     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

An algorithmic representation of fractional powers of positive operators

We are concerned with a new representation of fractional powers of operators by a series using exclusively contractive operators or operators with uniformly bounded powers. This representation is suitable for the construction of efficient approximations for which the error estimates are given and the convergence is investigated. The rate of convergence turns out to be exponential for bounded operators and polynomial for unbounded operators     

The representation of fractional powers of coercive differential operators

In this paper, under the definition of fractional powers by Straub, we will give the representation of fractional powers of coercive differential operators by using pseudo differential operators.