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5. Stability and instability of weighted composition operators

Author

Araujo, Jesus and Font, Juan J.

Subjects
Mathematics - Functional Analysis, Primary 47B38, Secondary 46J10, 47B33
Abstract

Let $\epsilon >0$. A continuous linear operator $T:C(X) \ra C(Y)$ is said to be {\em $\epsilon$-disjointness preserving} if $\vc (Tf)(Tg)\vd_{\infty} \le \epsilon$, whenever $f,g\in C(X)$ satisfy $\vc f\vd_{\infty} =\vc g\vd_{\infty} =1$ and $fg\equiv 0$. In this paper we address basically two main questions: 1.- How close there must be a weighted composition operator to a given $\epsilon$-disjointness preserving operator? 2.- How far can the set of weighted composition operators be from a given $\epsilon$-disjointness preserving operator? We address these two questions distinguishing among three cases: $X$ infinite, $X$ finite, and $Y$ a singleton ($\epsilon$-disjointness preserving functionals). We provide sharp stability and instability bounds for the three cases., Comment: 37 pages, 7 figures. A beamer presentation at http://www.araujo.tk

Published
2008

6. Isometric shifts and metric spaces

Author

Araujo, Jesus and Font, Juan J.

Subjects
Mathematics - Functional Analysis, Mathematics - General Topology, 47B38 (Primary) 54D65, 46J10 (Secondary)
Abstract

Let M be a complete metric space. It is proved that if the space or scalar-valued bounded continuous functions on M admits an isometric shift, then M is separable., Comment: 10 pages, LaTeX, no figures

Published
2001

23. Constructive Approximation of Continuous Interval-Valued Functions.

Author

Font, Juan J. and Macario, Sergio

Subjects
*CONTINUOUS functions, *HAUSDORFF spaces, *WEIERSTRASS-Stone theorem, *COMPACT spaces (Topology), *INTERVAL analysis
Abstract

In this paper we present a Stone-Weierstrass type result in the context of continuous interval-valued functions defined on a compact Hausdorff space. We also provide a Jackson type approximation result involving the modulus of continuity of interval-valued functions based on the gH-difference of intervals. Finally we show how an interval-valued continuous function can be approximated using interval neural networks. All the proofs are constructive. [ABSTRACT FROM AUTHOR]

Published
2022
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30. A version of Stone-Weierstrass theorem in Fuzzy Analysis

Author

Font, Juan J., Sanchis, Delia, and Sanchis, Manuel

Subjects
Fuzzy Analysis, Fuzzy functions, Fuzzy numbers
Abstract

[EN] Let C(K, E1) be the space of continuous functions defined between a compact Hausdorff space K and the space of fuzzy numbers E1 endowed with the supremum metric. We provide a sufficient set of conditions on a subspace of C(K, E1) in order that it be dense. We also obtain a similar result for interpolating families of C(K, E1)., This research is supported by Spanish Government (MTM2016-77143-P), Universitat Jaume I (Projecte P1-1B2014-35) and Generalitat Valenciana (Projecte AICO/2016/030).

Published
2017

32. Nonlinear diameter preserving maps on function spaces.

Author

Font, Juan J. and Hosseini, Maliheh

Subjects
*FUNCTION spaces, *FUNCTION algebras, *DIFFERENTIABLE functions, *DIAMETER, *DIFFERENTIABLE dynamical systems
Abstract

In this paper we study nonlinear diameter preserving mappings defined between function spaces and obtain generalizations of, basically, all known results concerning diameter preservers. In particular, we give a complete description for algebras of continuously differentiable functions, (little) Lipschitz algebras and dense function spaces. [ABSTRACT FROM AUTHOR]

Published
2020
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35. Constructive approximation of level continuous fuzzy functions1.

Author

Font, Juan J., Sanchis, Delia, and Sanchis, Manuel

Subjects
*WEIERSTRASS-Stone theorem, *FUZZY arithmetic, *FUNCTION spaces, *CONTINUOUS functions, *FUZZY numbers, *TOPOLOGY, *HAUSDORFF spaces
Abstract

We consider the space of continuous functions defined between a locally compact Hausdorff space and the space of fuzzy numbers endowed with the level convergence topology. We obtain a Stone-Weierstrass type theorem for such space of functions equipped with the compact open topology. As a corollary of the above results, we prove that such functions can be approximated by certain fuzzy-number-valued neural networks and sums of fuzzy-number-valued ridge functions. [ABSTRACT FROM AUTHOR]

Published
2019
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36. Constructive approximation of level continuous fuzzy functions1.

Author

Font, Juan J., Sanchis, Delia, and Sanchis, Manuel

Subjects
WEIERSTRASS-Stone theorem, FUZZY arithmetic, FUNCTION spaces, CONTINUOUS functions, FUZZY numbers, TOPOLOGY, HAUSDORFF spaces
Abstract

We consider the space of continuous functions defined between a locally compact Hausdorff space and the space of fuzzy numbers endowed with the level convergence topology. We obtain a Stone-Weierstrass type theorem for such space of functions equipped with the compact open topology. As a corollary of the above results, we prove that such functions can be approximated by certain fuzzy-number-valued neural networks and sums of fuzzy-number-valued ridge functions. [ABSTRACT FROM AUTHOR]

Published
2019
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38. Multilinear isometries on spaces of vector-valued continuous functions.

Author

Hosseini, Maliheh and Font, Juan J.

Subjects
*ISOMETRICS (Mathematics), *MULTILINEAR algebra, *VECTOR spaces, *CONTINUOUS functions, *VECTOR-valued measures, *ADDITIVE combinatorics
Abstract

In this paper we study multilinear isometries defined on certain subspaces of vector-valued continuous functions. We provide conditions under which such maps can be properly represented. Our results contain all known results concerning linear and bilinear isometries defined between spaces of continuous functions. The key result is a vector-valued version of the additive Bishop’s Lemma, which we think has interest in itself. [ABSTRACT FROM AUTHOR]

Published
2018
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40. Bilinear isometries on subspaces of continuous functions

Author

Font, Juan J. and Sanchis López, Manuel

Subjects
Funcions contínues, Subspaces of continuous functions, Functions, Continuous, Isometria (Matemàtica), Generalized peak point, Isometrics (Mathematics), Bilinear isometry
Abstract

Let A and B be strongly separating linear subspaces of C0(X) and C0(Y ), respectively, and assume that ∂A ̸= ∅ (∂A stands for the set of generalized peak points for A) and ∂B ≠ ∅. Let T : A×B −→ C0(Z) be a bilinear isometry. Then there exist a nonempty subset Z0 of Z, a surjective continuous mapping h : Z0 −→ ∂A × ∂B and a norm-one continuous function a : Z0 −→ K such that T(f,g)(z) = a(z)f(πx(h(z))g(πy(h(z)) for all z ∈ Z0 and every pair (f, g) ∈ A × B. These results can be applied, for example, to non-unital function algebras.

Published
2010
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43. Continuidad automática de operadores lineales y su representación como aplicaciones composición con peso

Author

Font, Juan J., Universitat Jaume I. Departament de Matemàtiques, Hernández Muñoz, Salvador, and Araujo Gómez, Jesús

Subjects
aplicaciones lineales, algebras de grupo, subespacios de funciones continuas, isometrias, aplicaciones separadoras
Abstract

En esta memoria se estudian básicamente dos tipos de operadores lineales: las aplicaciones separadoras y las isometrías.En el caso de las aplicaciones separadoras se analiza su continuidad automática en dos contextos: los espacios de funciones continuasy las álgebras de grupos localmante compactos y Abelianos. En ambos casos se representan las aplicaciones separadoras como aplicaciones composición con un peso.Por lo que respecta a las isometrías, su estudio se centra en su posible representación cuando están definidas sobre subespacios de funciones continuas.Se caracterizan los subespacios más pequeños sobre los que dicha representación es posible. Finalmente, se analizan las isometrías cuyo rango tiene codimensión finita.

Published
1996

48. Multilinear isometries on function algebras.

Author

Hosseini, Maliheh, Font, Juan J., and Sanchis, Manuel

Subjects
*ISOMETRICS (Mathematics), *MULTILINEAR algebra, *MATHEMATICAL functions, *SUBSPACES (Mathematics), *HAUSDORFF spaces
Abstract

Letbe function algebras (or more generally, dense subspaces of uniformly closed function algebras) on locally compact Hausdorff spaces, respectively, and letZbe a locally compact Hausdorff space. A-linear mapis called amultilinear (or k-linear) isometryifBased on a new version of the additive Bishop’s Lemma, we provide a weighted composition characterization of such maps. These results generalize the well-known Holsztyński’s theorem and the bilinear version of this theorem provided in Moreno and Rodríguez [Studia Math. 2005;166:83–91] by a different approach. [ABSTRACT FROM AUTHOR]

Published
2015
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