Maximal sectorial extensions and closed forms associated with them

We describe all closed sesquilinear forms associated with m-sectorial extensions of a densely defined sectorial operator with vertex at the origin.     

On maximal immediate extensions of valued fields

We study the maximal immediate extensions of valued fields whose residue fields are perfect and whose value groups are divisible by the residue characteristic if it is positive. In the case where there is such an extension which has finite transcendence degree we derive strong properties of the field and the extension and show that the maximal immediate extension is unique up to isomorphism, although these fields need not be Kaplansky fields. If the maximal immediate extension is an algebraic extension, we show that it is equal to the perfect hull and the completion of the field.     

The regular part of sectorial forms

We study the regular part of a densely defined sectorial form, first in the abstract setting and then under mild conditions for a differential sectorial form. The regular part of the latter turns out to be again a differential sectorial form. Moreover, we characterize when taking the real part of a differential sectorial form commutes with taking the regular part. An example shows that these two operations do not commute in general.     

On series of sectorial forms

We prove a convergence theorem for partial sums of sectorial forms with vertex zero and a common semi-angle. As an example, we prove an absorption theorem for sectorial forms.     

On the Extension of PM-Ring

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Extreme self‐adjoint extensions of a semibounded ‐difference operator

For a certain q‐difference operator introduced and studied in a series of articles by the same authors, we investigate its extreme self‐adjoint extensions, i.e., the so‐called Friedrichs and Kre?n extensions. We show that for the interval of parameters under consideration, the Friedrichs extension and the Kre?n extension are distinct and give values of the parameter in the von Neumann formulas that correspond to those extensions and describe their resolvent operators. A crucial rôle in our investigation plays the fact that both the Friedrichs and the Kre?n extensions are scale invariant.     

Convex extensions of the convex set valued maps

In this article the existence of the convex extension of convex set valued map is considered. Conditions are obtained, based on the notion of the derivative of set valued maps, which guarantee the existence of convex extension. The conditions are given, when the convex set valued map has no convex extension. The convex set valued map is specified, which is the maximal convex extension of the given convex set valued map and includes all other convex extensions. The connection between Lipschitz continuity and existence of convex extension of the given convex set valued map is studied.     

Essential extensions of Hausdorff spaces

In the category Haus of Hausdorff spaces the only injectives are the one-point spaces. Even though every Hausdorff spaceX has a maximal essential extension,X fails to have an injective hull, providedX has more than one point. A non-empty Hausdorff space has a proper essential extension if and only ifX is locally H-closed but not H-closed. In this case,X has (up to isomorphism) precisely one proper essential extension: the Obreanu-Porter extension (being simultaneously its maximal essential extension and its minimal H-closed extension). Completely parallel results hold for the categories SReg, Reg, and Tych of semi-regular, regular, and completely regular spaces respectively. In particular, the Alexandroff compactifications of locally compact, non-compact Hausdorff spaces are characterized categorically as the proper essential extensions of non-empty spaces in Tych (resp. Reg).Dedicated to my friend Nico Pumplün on his sixtieth birthday     

Embeddings of quasicells of iterative algebras

We study the relation between the projective and totally restricted extensions of preiterative algebras. We prove that each degree 1 projective extension of a quasicell of the algebra P _k ^* is a maximal subalgebra of a degree 1 totally restricted extension of the same quasicell. We show also that a projective extension of a quasicell can always be distinguished from its totally restricted extension in the same algebra by hyperidentities.     

Enveloping algebras of double Poisson-Ore extensions

It is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are preserved under iterated double Ore extensions are invariants of the Poisson enveloping algebra of a double Poisson-Ore extension.     

The regular part of second-order differential sectorial forms with lower-order terms

We present a formula for the regular part of a sectorial form that represents a general linear second-order differential expression that may include lower-order terms. The formula is given in terms of the original coefficients. It shows that the regular part is again a differential sectorial form and allows to characterise when also the singular part is sectorial. While this generalises earlier results on pure second-order differential expressions, it also shows that lower-order terms truly introduce new behaviour.     

Area integral functions for sectorial operators on L spaces

Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^infin functional calculus of sectorial operators on L^p-spaces hold true when the square functions are replaced by the area integral functions.     

A commutative diagram connected with a differential operator in a domain

A commutative diagram containing the maximal and minimal extensions of a differential operation in a domain and objects connected with them is obtained and the traces of functions from the kernel of the maximal extension are characterized.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1703–1709, December, 1991.     

Finite Normalizing Extensions of FSN Rings

Fully semiprimary Noetherian bimodules and their bimodule extensions are examined. In the presence of incomparability of the link graph of prime ideals, certain bimodule extensions preserve the fully semiprimary property. In particular, a finite normalizing extension ring of a fully semiprimary Noetherian ring is also fully semiprimary as a bimodule over the base ring. It is shown that the extension ring is itself a fully semiprimary ring. An application to crossed products over finite groups is given.     

The Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems

In this paper, the Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems is developed. A minimal and a maximal operators, GKN-sets, and a boundary space for the system are introduced. Algebraic characterizations of the domains of self-adjoint extensions of the minimal operator are given. A close relationship between the domains of self-adjoint extensions and the GKN-sets is established. It is shown that there exist one-to-one correspondences among the set of all the self-adjoint extensions, the set of all the d-dimensional Lagrangian subspaces of the boundary space, and the set of all the complete Lagrangian subspaces of the boundary space.     

Ore Extensions of Multiplier Hopf Algebras

The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed.     

Extensions of lattice-ordered groups

A number of conditions are specified which are sufficient for totally ordered groups with polycyclic factor group to contain a finite normal series of convex subgroups whose factors possess good enough properties. In any case studying such totally ordered groups is reduced to treating extensions of these groups as well as their virtually o-equivalent extensions. The concept of a virtually o-equivalent extension is a particular case of the notion of an Archimedean extension. Supported by RFBR project No. 03-01-00320. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 529–540, September–October, 2008.