Uniqueness of form extensions and domination of semigroups

In this article, we study uniqueness of form extensions in a rather general setting. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. Our main abstract result transfers uniqueness of form extension of a dominating form to that of a dominated form. This result can be applied to a multitude of examples including various magnetic Schrödinger forms on graphs and on manifolds.     

On solutions of differential inclusions in homogeneous spaces of functions

In this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions. These spaces include a large number of functional spaces like periodic functions and Bohr and Stepanov almost periodic functions. As an application, we consider some existence results for feedback control systems governed by degenerate differential equations of Sobolev type in a Banach space.     

Dissipative extension theory for linear relations

This work is devoted to dissipative extension theory for dissipative linear relations. We give a self-consistent theory of extensions by generalizing the theory on symmetric extensions of symmetric operators. Several results on the properties of dissipative relations are proven. Finally, we deal with the spectral properties of dissipative extensions of dissipative relations and provide results concerning particular realizations of this general setting.     

The Uniform Norming of Retractions on Short Intervals for Certain Function Spaces

For Lizorkin–Triebel spaces the family of extension operators is constructed which yield a minimal (in order) value of the norm among all possible extensions of a given function defined initially on the interval of an arbitrary small length.The techniques used restrict us to the one-dimensional case and spaces defined via differences of first order.     

Almost Periodic and Pseudo Almost Periodic Mild Solutions to a Partial Differential Equation VIA Fractional Operators

This article is mainly concerned with the existence and uniqueness of almost periodic and pseudo almost periodic mild solutions to a class of partial differential equations in Banach spaces. Under some reasonable assumptions, the main results are proved in α-norm. Our results cover the cases that the functions F ₁; F ₂ take values in different spaces such as X, X _α, and X _ß, where α ≤ β ∈ (0; 1).     

Fredholmness and invertibility of Toeplitz operators with matrix almost periodic symbols

We consider Toeplitz operators with symbols that are almost periodic matrix functions of several variables. It is shown that under certain conditions on the group generated by the Fourier support of the symbol, a Toeplitz operator is Fredholm if and only if it is invertible.

     

Rank-preserving extensions of band matrices

Existence and uniqueness problems are studied for extensions of a band matrix R such that the rank of the extension does not exceed the maximum of the ranks of the submatrices in the band of R. Applications are given to positive semidefinite extensions and extensions of lower-triangular matrices to contractions or to unitary matrices.     

The two straight line approach for periodic diffraction boundary-value problems

Boundary-transmission problems of first order for the Helmholtz equation are considered within the context of wave diffraction by a periodic strip grating and formulated as convolution type operators acting on a Bessel potential periodic space setting. Two boundary-value problems are studied for an arbitrary geometry of the grating: the oblique derivative and the classic Neumann boundary-value problems. The convolution type operators on the grating which correspond to the given boundary-transmission problems are associated with Toeplitz operators acting on spaces of matrix functions defined on composed contours. A Fredholm theory for periodic boundary-value problems of first order is established independently of the grating period and the Fredholm indices for the oblique derivative and the classic Neumann boundary-value problems are given.     

Function field genus theory for non-Kummer extensions

In this paper we first obtain the genus field of a finite abelian non-Kummer l–extension of a global rational function field. Then, using that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields and our previous results, we deduce the general expression of the genus field of a finite abelian extension of a global rational function field.     

On Cleft Extensions of the Tensor Product of Two Hopf Algebras

In this article, we first decompose a cleft extension for X ? Y into two cleft extensions for X and Y respectively, where X and Y are Hopf algebras on a commutative ring R. Conversely, we introduce the concept of consistent cleft Hopf extensions and prove that one can construct a cleft extension for X ? Y by two cleft extensions for X and Y if and only if these two cleft extensions are consistent. An example is also given to show an application of our main results.     

Norm closure and extension of the symbolic calculus for the cone algebra

We investigate the symbolic structure of an algebra of pseudodifferential operators on manifolds with conical singularities which has been introduced by B.-W. Schulze. Our main objective is the extension of the symbolic calculus of this algebra to its norm closure in an adapted scale of Sobolev spaces. This procedure yields Banach algebras and Fréchet algebras of singular integral operators with continuous principal symbols.     

On Minimal Extensions of Rings

Given two rings R ? S, S is said to be a minimal ring extension of R, if R is a maximal subring of S. In this article, we study minimal extensions of an arbitrary ring R, with particular focus on those possessing nonzero ideals that intersect R trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs &; Shapiro, and Ferrand &; Olivier, on commutative minimal extensions.     

Rank-preserving extensions of band matrices

Existence and uniqueness problems are studied for extensions of a band matrix R such that the rank of the extension does not exceed the maximum of the ranks of the submatrices in the band of R. Applications are given to positive semidefinite extensions and extensions of lower-triangular matrices to contractions or to unitary matrices.     

Abstract Korovkin-type theorems in modular spaces and applications

We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones.     

Spaces of piecewise-continuous almost-periodic functions and almost-periodic sets on the real line. II

The authors consider a series of spaces of piecewise-continuous almost periodic functions and study the properties of the elements of these spaces. The theory developed in the paper is then applied to investigate almost periodic linear pulse systems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 389–400, March, 1992.     

Mappings of higher order and nonlinear equations in some spaces of almost periodic functions

In the first part of the paper we examine mappings of higher order from a general point of view, that is, in normed spaces of bounded real-valued functions defined on R$\mathbb{R}$. Particular attention is paid to the relation of such mappings with the so-called autonomous superposition operators. Next we investigate mappings of higher order in Banach spaces of almost periodic functions and their perturbations. We also give necessary and sufficient conditions guaranteeing that a nonautonomous superposition operator acts in the space of almost periodic functions in the sense of Levitan and is uniformly continuous. In the Banach space of bounded almost periodic functions in the sense of Levitan we discuss mappings of higher order and a convolution operator. Some applications to nonlinear differential and integral equations are given.     

Almost periodic points and minimal sets in ω-regular spaces

In this paper a notion of ω-regular space is raised, which is an extension of that of regular space, and several known results concerning almost periodic points and minimal sets of maps are generalized from regular spaces to ω-regular spaces.