The Riesz–Kantorovich formula for lexicographically ordered spaces

Analysis and Stochastic

Product factorability of integral bilinear operators on Banach function spaces

[EN] This paper deals with bilinear operators acting in pairs of Banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices¿orthosymmetric maps, C¿-algebras¿zero product preserving operators, and classical and harmonic analysis¿integral bilinear operators. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties.The first author was supported by TUBITAK-The Scientific and Technological Research Council of Turkey, Grant No. 2211/E. The second author was supported by Ministerio de Economia y Competitividad (Spain) and FEDER, Grant MTM2016-77054-C2-1-P.Erdogan, E.; Sánchez Pérez, EA.; Gok, O. (2019). Product factorability of integral bilinear operators on Banach function spaces. Positivity. 23(3):671-696. https://doi.org/10.1007/s11117-018-0632-zS671696233Abramovich, Y.A., Kitover, A.K.: Inverses of Disjointness Preserving Operators. American Mathematical Society, Providence (2000)Abramovich, Y.A., Wickstead, A.W.: When each continuous operator is regular II. Indag. Math. (N.S.) 8(3), 281–294 (1997)Alaminos, J., Brešar, M., Extremera, J., Villena, A.R.: Maps preserving zero products. Studia Math. 193(2), 131–159 (2009)Alaminos, J., Brešar, M., Extremera, J., Villena, A.R.: On bilinear maps determined by rank one idempotents. Linear Algebra Appl. 432, 738–743 (2010)Alaminos, J., Extremera, J., Villena, A.R.: Orthogonality preserving linear maps on group algebras. Math. Proc. Camb. Philos. 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Shell Model in the Complex Energy Plane

This work reviews foundations and applications of the complex-energy continuum shell model that provides a consistent many-body description of bound states, resonances, and scattering states. The model can be considered a quasi-stationary open quantum system extension of the standard configuration interaction approach for well-bound (closed) systems.Comment: Topical Review, J. Phys. G, Nucl. Part. Phys, in press (2008

ON THE HYPERBOLICITY OF BASE SPACES FOR MAXIMALLY VARIATIONAL FAMILIES OF SMOOTH PROJECTIVE VARIETIES: Analytic Shafarevich hyperbolicity conjecture

This is the merger of the last version and the paper arXiv:1809.05891, with minor improvements.For smooth families with maximal variation, whose general fibers have semi-ample canonical bundle, the generalized Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by Popa-Schnell using the theory of Hodge modules and a theorem by Campana-P\u{a}un. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbolicity of moduli spaces of polarized manifolds with semi-ample canonical bundle. This answers a question by Viehweg-Zuo in 2003. We also prove the Kobayashi hyperbolicity of base spaces of effectively parametrized families of minimal projective manifolds of general type. This generalizes previous work by To-Yeung, in which they further assumed that these families are canonically polarized

Bands in partially ordered vector spaces with order unit

In an Archimedean directed partially ordered vector space X one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover Y of X. If X has an order unit, Y can be represented as C(?), where ? is a compact Hausdorff space. We characterize bands in X, and their disjoint complements, in terms of subsets of ?. We also analyze two methods to extend bands in X to C(?) and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by (1/4)2^(2^n) for n?2. We also construct examples of (n+1)-dimensional partially ordered vector spaces with (2n \choose n)+2 bands. This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when n?4