Approximation properties of tensor norms and operator ideals for Banach spaces

Ju Myung Kim

Type: Article

Publication Date: 2020-01-01

Citations: 2

DOI: https://doi.org/10.1515/math-2020-0116

Abstract

Abstract For a finitely generated tensor norm <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha , we investigate the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -approximation property ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -AP) and the bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -approximation property (bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda -bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> {\alpha }_{p,q} -AP <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo>≤</m:mo> <m:mi>∞</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>p</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>q</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:math> (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda -bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda -bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {g}_{p} -AP, then X has the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda -bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>w</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {w}_{p} -AP.

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Works Cited by This (12)

Action	Title	Year	Authors
+	Tensor Norms and Operator Ideals	1993	Andreas Defant Klaus Floret
+	The approximation property of order (p,q) in Banach spaces	1990	Juan Carlos Díaz Alcaide Juan Antonio López Molina María José Rivera Ortún
+	Introduction to Tensor Products of Banach Spaces	2002	Raymond A. Ryan
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+	Approximation properties of order <i>p</i> and the existence of non‐<i>p</i>‐nuclear operators with <i>p</i>‐nuclear second adjoints	1982	O. I. Reinov
+ PDF Chat	Sur certains produits tensoriels topologiques d'espaces de banach	1969	Mlle S. Chevet
+	Produits Tensoriels Topologiques Et Espaces Nucleaires	1966	Alexandre Grothendieck
+ PDF Chat	Applications à puissance nucléaire et applications de Hilbert-Schmidt dans les espaces de Banach	1966	Pierre Saphar
+	Absolutely Summing Operators	2006	Joseph Diestel Hans Jarchow Andrew Tonge
+ PDF Chat	Produits tensoriels topologiques et espaces nucléaires	1955	Alexandre Grothendieck