Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

Common invariant subspaces for collections of operators

Let be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that is finitely quasinilpotent at a vectorx ₀X whenever for any finite subset of the joint spectral radius of atx ₀ is equal 0. If such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant, subspace. If in addition, is a collection of positive operators on a Banach lattice, then has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia.     

Isometric representations of some quotients ofH ∞ of an annulus

Let denote an annulus,E a finite subset of with at least three elements, and the ideal of functions in which vanish at the points ofE. The quotient does not have a completely isometric representation on a finite dimensional Hilbert space. This complements a result of [11] which implies that the quotient has an isometric representation on a Hilbert space of dimension twice the cardinality ofE.     

Norm-principal bimodules of nest algebras II

Let be a nest in a separably acting factor . We present conditions for a factor to be a bimodule of the nest subalgebraalgß in which is singly generated in the norm topology. Some relevant results about the compact ideal , the quasitriangular algebra and the Calkin algebra are established.This work was supported, in part, by funds provided by the University of North Carolina at Charlotte.     

s-Prime elements in multiplicative lattices

Let be aC-lattice which is strong join principally generated. In this paper, we consider prime elements of for which every semiprimary element is primary. We show, for example, that a compact nonmaximal primep with this property is principal. We also show that if every primepm has this property, then is either a one dimensional domain or a primary lattice. It follows that if every primep satisfies the property, and if there are only a finite number of minimal primes in , then is the finite direct product of one-dimensional domains and primary lattices.     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

Commutator-finite orthomodular lattices

The class of orthomodular lattices which have only finitely many commutators is investigated. The following theorems are proved: contains the block-finite orthomodular lattices. Every irreducible element of is simple. Every element of is a direct product of a Boolean algebra and finitely many simple orthomodular lattices. The irreducible elements of which are modular, or are M-symmetric with at least one atom, have height two or less.     

Operators of fibered groups

Let (G, ) denote a finite groupG with fibration . The group Aut (G, ) of operators of (G, ) is closely related to the group of collineations of (G, ). In this paper we investigate the geometric properties imposed on (G, ) by requiring that Aut (G, )=AutG. We find that in many instances this algebraic property restricts the geometry to a very special form.Dedicated to Professor Helmut Karzel on the occasion of his 60th birthday.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

On Bergman-Toeplitz operators with commutative symbol algebras

Let be the unit disk in, be the Bergman space, consisting of all analytic functions from , and be the Bergman projection of onto . We constructC ^*-algebras , for functions of which the commutator of Toeplitz operators [T _a ,T _b ]=T _a T _b –T _b T _a is compact, and, at the same time, the semi-commutator [T _a ,T _b )=T _a T _b –T _ab is not compact.It is proved, that for each finite set =n ₀,n ₁, ...,n _m , where 1=n _{0 1} <... _m , andn _k {}, there are algebras of the above type, such that the symbol algebras Sym of Toeplitz operator algebras arecommutative, while the symbol algebras Sym of the algebras , generated by multiplication operators and , haveirreducible representations exactly of dimensions n ₀,n ₁,..., n _m .This work was partially supported by CONACYT Project 3114P-E9607, México.     

The Kreîn-Langer problem for Hilbert space operator valued functions on the band

We deal with the Kreîn-Langer problem for -valued functions on the band (–2a, 2a)×, where is the algebra of continuous linear operators on a Hilbert space ,a a finite positive number and a topological Abelian group. We show that every weakly continuous -indefinite function admits a strongly continuous -indefinite continuation to × with the same indefiniteness index . We give a parametrization of the extensions in terms of operator-valued Schur functions.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric -near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric -near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary -near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA N, then , with equality ifA if left invariant.     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

Series-parallel posets and relative Ockham lattices

A study is undertaken of order-reversing maps on series-parallel posets and structural characterisations are obtained of various subclasses of such ordered sets. The results are applied to complete the authors' earlier investigation of classes of finite relate lattices, where is a variety of Ockham lattices and the distributive lattice duals of the algebras in are required to be series-parallel.     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550     

A characterization of the line set of an odd-dimensional Baer subspace

Generalizing a theorem of Beutelspacher and Seeger, we consider line sets inP=PG(2t + 1,q),t IN, with the following properties: (1) any (t + 1)-dimensional subspace ofP contains at least one line of , (2) if a pointx ofP is incident with at least two lines of then the points in the factor geometryP/x which are induced by the lines of throughx form a blocking set of type (t, 1) inP/x, (3) any line of is coplanar with at least one further line of . We will show that the examples of minimal cardinality are exactly the line sets of Baer subspaces ofP.     

Partially stabilizing LQ-optimal control for stabilizable semigroup systems

The Linear-Quadratic optimal control problem with a partial stabilization constraint (LQPS) is considered for exponentially stabilizable infinite dimensional semigroup state-space systems with bounded sensing and control (having their transfer function with entries in the algebra . It is reported that the LQPS-optimal state-feedback operator is related to a nonnegative self-adjoint solution of an operator Riccati equation and it can be identified (1) by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper-stable transfer function algebra _, and (2) by obtaining any constant solution of a diophantine equation over _. These theoretical results are applied to a simple model of heat diffusion, leading to an approximation procedure converging exponentially fast to the LQPS-optimal state feedback operator.     

On a class of operators with finiterank self-commutators

This paper studies the class of pure operatorsA on a Hilbert space satisfying dimK _A <, where . The main tool is a pair of matrices and . A reproducing kernel Hilbert space model is introduced for a subclass of this class of operators. Some theorems are established for some subnormal operators as well as hyponormal operators in this class.