Essentially subnormal operators

An operator is essentially subnormal if its image in the Calkin algebra is subnormal. We shall characterize the essentially subnormal operators as those operators with an essentially normal extension. In fact, it is shown that an essentially subnormal operator has an extension of the form ``normal plus compact'.

The essential normal spectrum is defined and is used to characterize the essential isometries. It is shown that every essentially subnormal operator may be decomposed as the direct sum of a subnormal operator and some irreducible essentially subnormal operators. An essential version of Putnam's Inequality is proven for these operators. Also, it is shown that essential normality is a similarity invariant within the class of essentially subnormal operators. The class of essentially hyponormal operators is also briefly discussed and several examples of essentially subnormal operators are given.

     

Essentialité dans les bases additives

In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets P of a basis A such that A−P does not remains a basis. The existence of an essential subset for a basis is equivalent for this basis to be included, for almost all elements, in an arithmetic non-trivial progression. We show that for every basis A there exists an arithmetic progression with a biggest common difference containing A. Having this common difference a we are able to give an upper bound to the number of essential subsets of A: this is the radical's length of a (in particular there is always many finite essential subsets in a basis). In the case of essential subsets of cardinality 1 (essential elements) we introduce a way to “dessentialize” a basis. As an application, we definitively complete the result of Deschamps and Grekos about the majoration of essential elements of a basis by showing that for all basis A of order h, the number s of essential elements of A satisfy where , and we show that this inequality is best possible.     

Commutants modulo the compact operators of certain CSL algebras II

The commutant modulo compacts, or essential commutant, of a reflexive algebra with commutative subspace lattice is a C^* algebra which is the sum of the compact operators in L(H) and a C^* subalgebra of the core. We give a characterization of the essential commutant of a separably acting CSL algebra in terms of properties of the spectral measure of an operator in the intersection of the essential commutant and the core. This is used to determine some sufficient conditions on the lattice for when the essential commutant is norm generated by the projections it contains.     

Essential extensions, excellent extensions and finitely presented dimension

In Section 1 of this paper, we investigate the finitely presented dimension of an essential extension for a module, and obtain results concerning an essential extension of a torsion-free module. We partially answer the question: When is an essential extension of a finitely presented module (an almost finitely presented module) also finitely presented (almost finitely presented)? In Section 2, we study theC-excellent extensions and the finitely presented dimensions. We obtain some results on the homological dimensions of matrix rings and group rings.     

Essentially nilpotent rings

In this note we introduce a class of nil rings (called essentially nilpotent) which properly contains the class of nilpotent rings. A nil ring is said to be essentially nilpotent if it contains an essential right ideal which is nilpotent. Various properties of essentially nilpotent rings are investigated. A nil ring is essentially nilpotent if and only if it contains an essential right ideal which is leftT-nilpotent.     

An algorithm for finding the essential sets of arcs of certain graphs

Given a graph and a weight function which associates to each path an element of a Q-semiring, an essential set of arcs is defined as the complement of a maximal set of arcs which can be removed from the graph without changing the weight of the optimal paths for any pair of vertices. Conditions are given under which a graph admits a unique set of essential arcs and an algorithm is proposed to test for this condition and find this set of arcs.     

Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential

This paper is devoted to the study of essential self-adjointness of a relativistic Schrödinger operator with a singular homogeneous potential. From an explicit condition on the coefficient of the singular term, we provide a sufficient and necessary condition for essential self-adjointness.     

Strongly essential coalitions and the nucleolus of peer group games

Most of the known efficient algorithms designed to compute the nucleolus for special classes of balanced games are based on two facts: (i) in any balanced game, the coalitions which actually determine the nucleolus are essential; and (ii) all essential coalitions in any of the games in the class belong to a prespecified collection of size polynomial in the number of players. We consider a subclass of essential coalitions, called strongly essential coalitions, and show that in any game, the collection of strongly essential coalitions contains all the coalitions which actually determine the core, and in case the core is not empty, the nucleolus and the kernelcore. As an application, we consider peer group games, and show that they admit at most 2n−1 strongly essential coalitions, whereas the number of essential coalitions could be as much as 2ⁿ⁻¹. We propose an algorithm that computes the nucleolus of an n-player peer group game in time directly from the data of the underlying peer group situation.Research supported in part by OTKA grant T030945. The authors thank a referee and the editor for their suggestions on how to improve the presentation     

On essential sets and essential components of efficient solutions for vector optimization problems

In this paper, we first introduce the notions of an essential set and an essential component of the set of efficient solutions for continuous vector optimizations on a nonempty compact subset of a metric space. Then we show that for each of these vector optimizations, each set of all efficient solutions corresponding to the same optimal values is essential. Basing on this result, we give full characterizations of an essential point, an essential set and an essential component, respectively. As an application, we prove that for continuous quasiconvex vector optimization problems on a nonempty compact subset of a metric vector space, each component of the set of efficient solutions is essential even though the efficient solution set is not connected.     

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     

Decision problems in the space of Dehn fillings

Normal surface theory is used to study Dehn fillings of a knot-manifold. We use that any triangulation of a knot-manifold may be modified to a triangulation having just one vertex in the boundary. In this situation, it is shown that there is a finite computable set of slopes on the boundary of the knot-manifold, which come from boundary slopes of normal or almost normal surfaces. This is combined with existence theorems for normal and almost normal surfaces to construct algorithms to determine precisely those manifolds obtained by Dehn filling of a given knot-manifold that: (1) are reducible, (2) contain two-sided incompressible surfaces, (3) are Haken, (4) fiber over S¹, (5) are the 3-sphere, and (6) are a lens space. Each of these algorithms is a finite computation.Moreover, in the case of essential surfaces, we show that the topology of each filled manifold is strongly reflected in the combinatorial properties of a triangulation of the knot-manifold with just one vertex in the boundary. If a filled manifold contains an essential surface then the knot-manifold contains an essential normal vertex solution which caps off to an essential surface of the same type in the filled manifold. (Normal vertex solutions are the premier class of normal surface and are computable.)     

On right congruences of semigroups having no proper essential right congruences

A right congruence ?? in a semigroup S is essential if for any right congruence ?? we have ??????=?? (the identity relation) implies ??=??. Clearly, the universal relation, ??, is an essential right congruence. We say ?? is proper if ??????. In this paper we get a necessary and sufficient condition for a semigroup with an identity element?1 and having no proper essential right congruences to have a distributive lattice of right congruences.     

Algorithms for finding proper essential surfaces in 3-manifolds

In this paper, we present an algorithm which, for a given compact orientable irreducible boundary irreducible 3-manifold M, verifies whether M contains an essential orientable surface (possibly, with boundary), whose genus is at most N. The algorithm is based on Haken’s theory of normal surfaces, and on a trick suggested by Jaco and consisting in estimating the mean length of boundary curves in an unknown essential surface of a given genus in the given manifold.     

Exclusive and essential sets of implicates of Boolean functions

In this paper we study relationships between CNF representations of a given Boolean function f and certain sets of implicates of f. We introduce two definitions of sets of implicates which are both based on the properties of resolution. The first type of sets, called exclusive sets of implicates, is shown to have a functional property useful for decompositions. The second type of sets, called essential sets of implicates, is proved to possess an orthogonality property, which implies that every CNF representation and every essential set must intersect. The latter property then leads to an interesting question, to which we give an affirmative answer for some special subclasses of Horn Boolean functions.     

Robust Solution of Nonconvex Global Optimization Problems

The concept of ɛ-approximate optimal solution as widely used in nonconvex global optimization is not quite adequate, because such a point may correspond to an objective function value far from the true optimal value, while being infeasible. We introduce a concept of essential ɛ-optimal solution, which gives a more appropriate approximate optimal solution, while being stable under small perturbations of the constraints. A general method for finding an essential ɛ-optimal solution in finitely many steps is proposed which can be applied to d.c. programming and monotonic optimization.     

Immersed essential surfaces and Dehn surgery

It is known that an embedded essential surface F in a hyperbolic manifold M remains essential in Dehn filling spaces M(γ) for most slopes γ on a torus boundary component T of M. The main theorem of this paper is to generalize this result to immersed surfaces. More explicitly, if an immersed essential surface F has coannular slopes β₁,…,β_n on T, then there is a constant K such that F remains essential in M(γ) when Δ(γ,β_i)>K for all i. It will also be shown that all but finitely many Freedman tubings of a geometrically finite surface in M are π₁-injective.     

A hyperstructural approach to essentiality

Let M be a hypermodule over a hyperring R such that the intersection of any two subhypermodules of M, is a subhypermodule of M. We introduce the concept of an essential subhypermodule in M relative to an arbitrary subhypermodule T of M, which is called a T-essential subhypermodule of M. Our main goal in this work is to investigate properties of (relative) essential subhypermodules. We apply this concept to introduce extending hypermodules. Examples are provided to illustrate different concepts.     

Essential Closures and AC Spectra for Reflectionless CMV,Jacobi, and Schrödinger Operators Revisited

We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures.As an elementary application of the notion of the essential closure of subsets of ? we revisit the fact that CMV, Jacobi, and Schrödinger operators, reflectionless on a set ? of positive Lebesgue measure, have absolutely continuous spectrum on the essential closure \({\overline{\mathcal{E}}}^{e}\) of the set ? (with uniform multiplicity two on ?). Though this result in the case of Schrödinger and Jacobi operators is known to experts, we feel it nicely illustrates the concept and usefulness of essential closures in the spectral theory of classes of reflectionless differential and difference operators.     

Matrices and graphs of essential dependence of proper families of functions

This paper considers proper families of functions, which are used in functional specification of Latin squares of large size over the set of n-dimensional binary vectors. Proper families of functions are studied from the viewpoint of the intrinsic structure of the corresponding graphs of essential dependence and their adjacency matrices. Various necessary and sufficient conditions for a binary matrix to be treated as the adjacency matrix of the graph of essential dependence of a proper family of functions are derived. Also, transformations of matrices are considered under which the indicated property is preserved. It is demonstrated that any directed graph without loops and multiple edges can be embedded as an induced subgraph into the graph of essential dependence of some proper family of functions. Moreover, such embedding is reasonably economical, and the functions of the resulting proper family inherit properties of the functions that realize the original graph as the graph of essential dependence.     

On nilpotent ideals in the cohomology ring of a finite group

In this paper we find upper bounds for the nilpotency degree of some ideals in the cohomology ring of a finite group by studying fixed point free actions of the group on suitable spaces. The ideals we study are the kernels of restriction maps to certain collections of proper subgroups. We recover the Quillen-Venkov lemma and the Quillen F-injectivity theorem as corollaries, and discuss some generalizations and further applications.We then consider the essential cohomology conjecture, and show that it is related to group actions on connected graphs. We discuss an obstruction for constructing a fixed point free action of a group on a connected graph with zero “k-invariant” and study the class related to this obstruction. It turns out that this class is a “universal essential class” for the group and controls many questions about the groups essential cohomology and transfers from proper subgroups.