Prime modules of a gamma ring

We define a prime ΓM-module for a Γ-ringM. It is shown that a subsetP ofM is a prime ideal ofM if and only ifP is the annihilator of some prime ΓM-moduleG. s-prime ideals ofM were defined by the first author. We defines-modules ofM, analogous to a concept defined by De Wet for rings. It is shown that a subsetQ ofM is ans-prime ideal ofM if and only ifQ is the annihilator of somes-moduleG ofM. Relationships between prime ΓM-modules and primeR-modules are established, whereR is the right operator ring ofM. Similar results are obtained fors-modules.     

Strong semi-simplicity

Dedicated to the memory of Ferenc Szász     

Unitary couplings,scattering suboperators and regular factorizations of bounded operator‐valued functions III

The present part of the paper develops a geometrical approach to the study of factorizations of contractive operator functions in the class . The proposed approach is based on the methods of the theory of unitary couplings elaborated in the previous sections of this article. This technique is related, first of all, to the concepts of a factorizing channel of a coupling and the factorization of the scattering suboperator of this coupling that it generates.     

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils

Self-adjoint quadratic operator pencilsL()= ² A + B + C with a noninvertible leading operatorA are considered. In particular, a characterization of the spectral points of positive and of negative type ofL is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.     

On finitely generated projective modules and multiplication modules

This paper is based on the M. Sc. thesis written by the third author under the supervision of the first two authors. It was submitted to the University of Baghdad in January 1987.     

The dual of a finitely generated multiplication module

This paper is based on the M. Sc. thesis written by the third author under the supervision of the first two authors. It was submitted to the University of Baghdad in 1986.     

Sums and extensions of vector lattice homomorphisms

This paper gives a summary account, with minimal or no proofs, first of some results which characterize order bounded linear operators which are sums of lattice homomorphisms, or more generally of orthomorphisms; and secondly of theorems concerning extensions of vector lattice homomorphisms (theorems of Hahn-Banach type if you will). In all cases we assume that domain and range are vector lattices and that the range is Dedekind complete. The results vary from historical (pre-1940) to recent (1990). The most recent work, on sums of lattice homomorphisms, is covered in §1 and the more classical work on extension theorems is dealt with in §2.     

Existence and extensions of positive linear operators

In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators. In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators. In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied. Dedicated to Prof. Romulus Cristescu on his 80th birthday     

Commutant lifting theorem and interpolation in discrete nest algebras

Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.     

Eigenvalue extensions of Bohr’s inequality

We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohr’s inequality due to Vasi? and Ke?ki?.     

A proof of Parrott's theorem on quotient norms

We prove, as an application of our positive extension argument, a theorem of Parrott concerning the quotient norm with respect to spaces of Hilbert space operators.