Eccentric <i>p</i>-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs
The extension of the concept of <i>p</i>-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space <i>M</i> afforded by the associate...
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2024-11-01
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| author | Roger Arnau Enrique A. Sánchez Pérez Sergi Sanjuan |
| author_facet | Roger Arnau Enrique A. Sánchez Pérez Sergi Sanjuan |
| author_sort | Roger Arnau |
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| description | The extension of the concept of <i>p</i>-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space <i>M</i> afforded by the associated Arens–Eells space, along with the duality between <i>M</i> and the metric dual space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mo>#</mo></msup></semantics></math></inline-formula> defined by the real-valued Lipschitz functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>.</mo></mrow></semantics></math></inline-formula> However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of <i>p</i>-summability) exist. One approach involves considering specific subsets of the unit ball of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mo>#</mo></msup></semantics></math></inline-formula> for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry. |
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| spelling | doaj-art-53a305d9ac6a4ad8b06ed482f151300b2024-11-26T17:50:49ZengMDPI AGAxioms2075-16802024-11-01131176010.3390/axioms13110760Eccentric <i>p</i>-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and GraphsRoger Arnau0Enrique A. Sánchez Pérez1Sergi Sanjuan2Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, SpainInstituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, SpainInstituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, SpainThe extension of the concept of <i>p</i>-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space <i>M</i> afforded by the associated Arens–Eells space, along with the duality between <i>M</i> and the metric dual space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mo>#</mo></msup></semantics></math></inline-formula> defined by the real-valued Lipschitz functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>.</mo></mrow></semantics></math></inline-formula> However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of <i>p</i>-summability) exist. One approach involves considering specific subsets of the unit ball of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mo>#</mo></msup></semantics></math></inline-formula> for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.https://www.mdpi.com/2075-1680/13/11/760Lipschitzmetricsummabilityintegral inequalitiesdomination |
| spellingShingle | Roger Arnau Enrique A. Sánchez Pérez Sergi Sanjuan Eccentric <i>p</i>-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs Axioms Lipschitz metric summability integral inequalities domination |
| title | Eccentric <i>p</i>-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs |
| title_full | Eccentric <i>p</i>-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs |
| title_fullStr | Eccentric <i>p</i>-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs |
| title_full_unstemmed | Eccentric <i>p</i>-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs |
| title_short | Eccentric <i>p</i>-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs |
| title_sort | eccentric i p i summing lipschitz operators and integral inequalities on metric spaces and graphs |
| topic | Lipschitz metric summability integral inequalities domination |
| url | https://www.mdpi.com/2075-1680/13/11/760 |
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