An Abstract Version of a Result of Fong and Sucheston

1978 ◽  
Vol 21 (2) ◽  
pp. 247-248
Author(s):  
P. E. Kopp
Keyword(s):  
General Result ◽  
Representation Theorem ◽  
Analytic Proof ◽  
Abstract Version ◽  
Operator Version

Nagel [3] has given a purely functional-analytic proof of Akcoglu and Sucheston's operator version [1] of the Blum-Hanson theorem. The purpose of this note is to show that the same techniques may be applied to obtain a proof, in the context of (AL)-spaces, of a more general result due to Fong and Sucheston [2]. By Kakutani's representation theorem, any (AL)-space can of course be represented as an L-1-space. Thus the present result is simply a reformulation of that of Fong and Sucheston.

scholarly journals Closure of the Cone of Sums of 2d-powers in Certain Weighted ℓ1-seminorm Topologies

2014 ◽  
Vol 57 (2) ◽  
pp. 289-302 ◽  
Author(s):  
Mehdi Ghasemi ◽  
Murray Marshall ◽  
Sven Wagner
Keyword(s):  
Commutative Semigroup ◽  
Polynomial Ring ◽  
General Result ◽  
Representation Theorem ◽  
Sums Of Squares ◽  
Absolute Value

AbstractIn a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares in the polynomial ring in the topology induced by the ℓ1-norm is equal to Pos([–1; 1]n), the cone consisting of all polynomials that are non-negative on the hypercube [–1,1]n. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted -seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theoremfrom2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of 2d-powers, proving, in particular, that for any integer d ≥ 1, the closure of the cone of sums of 2d-powers in the topology induced by the -norm is equal to Pos([–1; 1]n).

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

scholarly journals Study of Two Families of Generalized Yager’s Implications for Describing the Structure of Generalized (h,e)-Implications

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1490
Author(s):  
Raquel Fernandez-Peralta ◽  
Sebastia Massanet ◽  
Arnau Mir
Keyword(s):  
Representation Theorem ◽  
Internal Factors ◽  
Fuzzy Implication ◽  
The Family

In this study, we analyze the family of generalized (h,e)-implications. We determine when this family fulfills some of the main additional properties of fuzzy implication functions and we obtain a representation theorem that describes the structure of a generalized (h,e)-implication in terms of two families of fuzzy implication functions. These two families can be interpreted as particular cases of the (f,g) and (g,f)-implications, which are two families of fuzzy implication functions that generalize the well-known f and g-generated implications proposed by Yager through a generalization of the internal factors x and 1x, respectively. The behavior and additional properties of these two families are also studied in detail.

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

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