An analyticity criterion for regularized semigroups

A generalization of Kato's analyticity criterion for -semigroups to exponentially bounded regularized semigroups is given by using the method of Laplace transforms.

     

On one criterion for analyticity of functions

We prove a generalization of the well-known Dzyadyk theorem that gives an interesting geometric criterion for the analyticity of functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1410–1418, October, 2007.     

Criteria for analyticity of subordinate semigroups

Let ψ be a Bernstein function. A. Carasso and T. Kato obtained necessary and sufficient conditions for ψ to have the property that ψ(A) generates a quasibounded holomorphic semigroup for every generator A of a bounded C ₀-semigroup in a Banach space, in terms of some convolution semigroup of measures associated with ψ. We give an alternative to Carasso-Kato’s criterion, and derive several sufficient conditions for ψ to have the above-mentioned property. The author was supported in part by the State Program of Fundamental Research of Republic of Belarus under the contract number 20061473.     

On partial analyticity of CR mappings

We study the problem of holomorphic extension of a smooth CR mapping from a real analytic hypersurface to a real algebraic set in complex spaces of different dimensions. Received November 26, 1998; in final form March 23, 1999 / Published online October 11, 2000     

On product formulas for nonlinear semigroups

We study a number of sufficient conditions which guarantee the convergence of semigroup product formulas of the type

$\text{H}{}_{\mathrm{t}}\text{=}\text{lim}\text{n→∞}\text{(F}{}_{\mathrm{<mtext>t</mtext><mtext>n</mtext>}}\text{°G}{}_{\mathrm{<mtext>t</mtext><mtext>n</mtext>}}\text{)}{}^{\mathrm{n}}$

and its generalizations. Our hypotheses differ from those of other authors in that we do not assume in advance that the limit operator is a generator. Rather this is a consequence and hence the above formula yields an existence theorem (local in time) for nonlinear semigroups. A number of smoothness properties are studied as well. The results may be applied to and are motivated by the Navier-Stokes equations.     

On U-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semi-groups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in HU on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup WU and a V-ample semigroup ...     

On Ehresmann semigroups

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

     

On implemented semigroups

We shall be concerned with implemented semigroups of the form U(t)X := T(t)XS(t) for X ∈ L(F,E) and t ≥ 0 , where (T(t)) _{t ≥ 0} and (S(t)) _{t ≥ 0} are C ₀ -semigroups on the Banach spaces E and F , respectively. The aim of this article is to clarify the structure of this semigroup and its ``generator' with the help of inter- and extrapolation techniques. Moreover, using positivity arguments, we present a short proof of the infinite dimensional Liapunov Stability Theorem on Hilbert spaces as an application of our results. May 15, 2000     

On exclusive semigroups

The following problem was posed by the second author in ‘Semigroup Forum’, Vol. 1, No. 1, 1970, p. 91: Describe the structure of semigroups S such that for all a, b, c ε S, abc=ab, ac or bc.     

On Gopalsamy and Liu's conjecture for global stability in a population model

For a logistic equation with a piecewise constant argument which models the dynamics of a population of a single species undergoing a density-dependent harvesting, Gopalsamy and Liu (J. Math. Anal. Appl. 224 (1998) 59–80), have offered a condition of the growth rate of species and conjectured that this is a necessary and sufficient condition of not only the asymptotic stability but also the global asymptotic stability for the positive equilibrium of the equation. But until now, there were no mathematical answers except computer simulations.In this paper, we establish a mathematically rigorous proof of a partial affirmative answer of this conjecture.