Norms of basic elementary operators on algebras of regular operators

We show that if E is an atomic Banach lattice with an ordercontinuous norm, A, B ∈ L^r(E) and M_A,_B is the operator on L^r(E) defined by M_A,_B(T) = AT B then ||M_A,_B||r = ||A||_r||B||_r but that there is no real α > 0 such that ||MA,B || ≥ α ||A||r||B ||r.

A Study of Elementary Operators

Department of Mathematics, Cochin University of Science and Technology

Spectrally isometric elementary operators

We present criteria for unital elementary operators (of small length) on unital semisimple Banach algebras to be spectral isometries. The surjective ones among them turn out to be algebra automorphisms.

More elementary operators that are spectrally bounded

We discuss some necessary and some sufficient conditions for an elementary operator x↦∑ⁿ_i=1a_ixb_i on a Banach algebra A to be spectrally bounded. In the case of length three, we obtain a complete characterisation when A acts irreducibly on a Banach space of dimension greater than three.

Locally quasi-nilpotent elementary operators

Let A be a unital dense algebra of linear mappings on a complex vector space X. Let φ = Σⁿ _i=1 M_ai,_bi be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a1, . . . , an} is locally linearly independent, then the local dimension of V (φ) = span{b_ia_j : 1 ≤ i, j ≤ n} is at most n(n−1) 2 . If ldim V (φ) = n(n−1) 2 , then there exists a representation of φ as φ = Σⁿ _i=1 M_ui,_vi with v_iu_j = 0 for i ≥ j. Moreover, we give a complete characterization of locally quasinilpotent elementary operators of length 3.

Elementary operators-still not elementary?

Properties of elementary operators, that is, finite sums of two-sided multiplications on a Banach algebra, have been studied under a vast variety of aspects by numerous authors. In this paper we review recent advances in a new direction that seems not to have been explored before: the question when an elementary operator is spectrally bounded or spectrally isometric. As with other investigations, a number of subtleties occur which show that elementary operators are still not elementary to handle.

Uma Fundamentação Intervalar Aplicada à Morfologia Matemática

This work present a interval approach to deal with images with that contain uncertainties, as well, as treating these uncertainties through morphologic operations. Had been presented two intervals models. For the first, is introduced an algebraic space with three values, that was constructed based in the tri-valorada logic of Lukasiewiecz. With this algebraic structure, the theory of the interval binary images, that extends the classic binary model with the inclusion of the uncertainty information, was introduced. The same one can be applied to represent certain binary images with uncertainty in pixels, that it was originated, for example, during the process of the acquisition of the image. The lattice structure of these images, allow the definition of the morphologic operators, where the uncertainties are treated locally. The second model, extend the classic model to the images in gray levels, where the functions that represent these images are mapping in a finite set of interval values. The algebraic structure belong the complete lattices class, what also it allow the definition of the elementary operators of the mathematical morphology, dilation and erosion for this images. Thus, it is established a interval theory applied to the mathematical morphology to deal with problems of uncertainties in images

Full and hat inductive definitions are equivalent in NBG

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA 0, those in second order set theory extending NBG (as well as those in n + 3-th order number theory extending the so-called Bernays−Gödel expansion of full n + 2-order number theory etc.) are. In this article, we establish the equivalence between Δ10\bf-LFP and Δ10\bf-FP, which assert the existence of a least and of a (not necessarily least) fixed point, respectively, for positive elementary operators (or between Δn+20\bf-LFP and Δn+20\bf-FP). Our proof also shows the equivalence between ID 1 and ^ID1, both of which are defined in the standard way but with the starting theory PA replaced by ZFC (or full n + 2-th order number theory with global well-ordering).

Mapping of composite hadrons into elementary hadrons and effective hadronic Hamiltonians

A mapping technique is used to derive in the context of constituent quark models effective Hamiltonians that involve explicit hadron degrees of freedom. The technique is based on the ideas of mapping between physical and ideal Fock spaces and shares similarities with the quasiparticle method of Weinberg. Starting with the Fock-space representation of single-hadron states, a change of representation is implemented by a unitary transformation such that composites are redescribed by elementary Bose and Fermi field operators in an extended Fock space. When the unitary transformation is applied to the microscopic quark Hamiltonian, effective, hermitian Hamiltonians with a clear physical interpretation are obtained. Applications and comparisons with other composite-particle formalisms of the recent literature are made using the nonrelativistic quark model. (C) 1998 Academic Press.

On The Cauchy Problem for Non Effectively Hyperbolic Operators, The Ivrii-Petkov-Hörmander Condition and the Gevrey Well Posedness

2000 Mathematics Subject Classification: 35L15, Secondary 35L30.