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, )C(X, ) and a continuous function :X[0, 1], it is shown that under suitable assumptions there exists a positive contraction semigroup onC(X, ) that can be represented in terms of the Lototsky-Schnabl operators associated withT and . Several properties of this semigroup are investigated. In particular, its infinitesimal generator is determined in a core of its domain. WhenX ^p for somep1, then the generator is shown to be a degenerate elliptic second order differential operator.Dedicated to Professor George Maltese on the occasion of his 60^th birthday     

On a class of t-Archimedean semigroups

Communicated by H.-J. Hoehnke     

On one class of differential operators and their application

We use a generalized differentiation operator to construct a generalized shift operator, which makes it possible to define a generalized convolution operator in the space H(?). Next, we consider the characteristic function of this operator and introduce a generalized Laplace transform. We study the homogeneous equation of the generalized convolution operator, investigate its solvability, and consider the multipoint Vallée Poussin problem.     

On a class of selfadjoint operators in Krein space and their compact perturbations

For a class of selfadjoint operators in a Krein space containing the definitizable selfadjoint operators a funetional calculus and the spectral function are studied. Stability properties of the spectral function with respect to small compact perturbations of the resolvent are proved.     

On a class of limit distributions

One establishes the relationship between the asymptotic behavior as x of the distribution function F(x) from the class of the mixtures of normal distributions and the asymptotic behavior of the distribution function G(x).Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 40–48, 1986.     

On a class of quasi-Fredholm operators

We study a class of bounded linear operators acting on a Banach spaceX called B-Fredholm operators. Among other things we characterize a B-Fredholm operator as the direct sum of a nilpotent operator and a Fredholm operator and we prove a spectral mapping theorem for B-Fredholm operators.IMemory of my father, Sidi-Bouhouria 1914-0991.     

On reducibility of semigroups of compact quasinilpotent operators

The following generalization of Lomonosov's invariant subspace theorem is proved. Let be a multiplicative semigroup of compact operators on a Banach space such that for every finite subset of , where denotes the Rota-Strang spectral radius. Then is reducible.

This result implies that the following assertions are equivalent:

(A) For each infinite-dimensional complex Hilbert space , every semigroup of compact quasinilpotent operators on is reducible.

(B) For every complex Hilbert space , for every semigroup of compact quasinilpotent operators on , and for every finite subset of it holds that .

The question whether the assertion (A) is true was considered by Nordgren, Radjavi and Rosenthal in 1984, and it seems to be still open.

     

On a class of coupled Hamiltonian operators and their integrable hierarchies with two potentials

We discuss at first in this paper the Gauge equivalence among several u‐linear Hamiltonian operators and present explicitly the associated Gauge transformation of Bäcklund type among them. We then establish the sufficient and necessary conditions for the linear superposition of the discussed u‐linear operators and matrix differential operators with constant coefficients of arbitrary order to be Hamiltonian, which interestingly shows that the resulting Hamiltonian operators survive only up to the third differential order. Finally, we explore a few illustrative examples of integrable hierarchies from Hamiltonian pairs embedded in the resulting Hamiltonian operators.     

On a limit class of approximation spaces

We describe the behaviour under interpolation of a limit class of approximation spaces. We characterize them as extrapolation spaces. Moreover, we study the boundedness of certain operators on these spaces. As an application, we derive several results on Macaev operator ideals.