Dynamic effect algebras

We introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.     

Resonant Delocalization on the Bethe Strip

Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schrödinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.     

Idempotent Block Splitting on Partial Partitions,I: Isotone Operators

Image segmentation algorithms can be modelled as image-guided operators (maps) on the complete lattice of partitions of space, or on the one of partial partitions (i.e., partitions of subsets of the space). In particular region-splitting segmentation algorithms correspond to block splitting operators on the lattice of partial partitions, in other words anti-extensive operators that act by splitting each block independently. This first paper studies in detail block splitting operators and their lattice-theoretical and monoid properties; in particular we consider their idempotence (a requirement in image segmentation). We characterize block splitting openings (kernel operators) as operators splitting each block into its connected components according to a partial connection; furthermore, block splitting openings constitute a complete sublattice of the complete lattice of all openings on partial partitions. Our results underlie the connective approach to image segmentation introduced by Serra. The second paper will study two classes of non-isotone idempotent block splitting operators, that are also relevant to image segmentation.     

Algebraic axiomatization of tense intuitionistic logic

We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.     

Incompleteness of the linear span of the positive compact operators

We show that even in the case of a Banach lattice with an order continuous norm (or whose dual has an order continuous norm) the linear span of the positive compact operators on need not be complete under the regular norm.

     

On the lattice structure of kernel operators

Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a sublattice that is lattice isomorphic to the space of transition kernels. As an application we present a purely analytic proof of Doob's theorem concerning stability of transition semigroups.     

Asymmetric fermi surfaces for magnetic schrödinger operators

We consider Schrödinger operators with periodic magnetic field having zero flux through a fundamental cell of the period lattice. We show that, for a generic small magnetic field and a generic small Fermi energy, the corresponding Fermi surface is convex and not invariant under inversion in any point.     

Pieri’s formula for generalized Schur polynomials

Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutation relation of generalized Schur operators implies Pieri's formula for generalized Schur polynomials.     

A note on the generalized symmetries of lattice hierarchies

We show that the recursion operators of the integrable lattice equations usually considered in the literature can also be used to generate hierarchies of differential-delay equations. All members of these hierarchies of lattice and differential-delay equations commute. It is thus seen that differential-delay hierarchies provide a broader context within which to place lattice hierarchies.     

Lattice polytopes,Hecke operators,and the Ehrhart polynomial

Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.      

Green’s formula for difference operators

We construct an analogue of the classic Green’s formula for linear partial differential operators for difference operators on a multidimensional integer lattice.     

Positive Absolutely Summing Operators on the Köthe Spaces

In this paper we characterize the positive absolutely summing operators on the Köthe space E(X), with X a Banach lattice, extending a previous result. We prove that a composition operator of two positive absolutely summing operators is a positive absolutely summing operator. An interpolation result for the positive absolutely summing operators is obtained.     

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

     

Einstein 4-manifolds and nonpositive isotropic curvature

We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach lattice under the regular norm, for that Banach lattice to have an order continuous norm, to be an AL-space or an AM-space.     

Two dimensional unital Riesz algebras,their representations and norms

We describe all two dimensional unital Riesz algebras and study representations of them in Riesz algebras of regular operators. Although our results are not complete, we do demonstrate that very varied behaviour can occur even though all these algebras can be given a Banach lattice algebra norm.     

Elementary operators and the Aluthge transform

We characterize essential normality for certain elementary operators acting on the Hilbert-Schmidt class. We find the Aluthge transform of an elementary operator of length one. We show that the Aluthge transform of an elementary 2-isometry need not be a 2-isometry. We also characterize hermitian elementary operators of length 2.     

The lattice of closure operators on a subgroup lattice

We say a lattice L is a subgroup lattice if there exists a group G such that Sub(G)?L, where Sub(G) is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of closure operators which act on the subgroup lattice of a finite group G is itself a subgroup lattice if and only if G is cyclic of prime power order.     

About positive weak Dunford-Pettis operators on Banach lattices

We characterize Banach lattices for which each positive weak Dunford-Pettis operator from a Banach lattice into another dual Banach lattice is almost Dunford-Pettis. Also, we give some sufficient and necessary conditions for which the class of positive weak Dunford-Pettis operators coincides with that of positive Dunford-Pettis operators, and we derive some consequences.     

Identities of arithmetic type between values of the theta function associated to a lattice in ℝ d and its derivatives

In this paper, we introduce a notion of similarly self dual lattice in a d-dimensional Euclidean space and a classical Jacobi theta function is associated to such a lattice. We establish identities of arithmetic type between values of this theta function and its successive derivatives. This work can be related to the spectral theory of the Landau operators.      

Perturbation of regular operators and the order essential spectrum

For regular operators on a Banach lattice, we introduce and investigate two notions of order essential spectrum analogous to the essential spectrum and the Weyl spectrum for operators on Banach spaces. We also discuss related questions on the behaviour of the order spectrum under perturbation by r-compact operators.