Ideal topologies and corresponding approximation properties

Abstract

Abstract. We propose a unifying approach to numerous approximation properties in Banach spaces studied from the 1930s up to our days. To do so, we introduce the concept of ideal topology and say that a Banach space E has the (I, J , τ )-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the ideal topology τ , by operators belonging to the operator ideal J . This concept recovers many classical/recent approximation properties as particular instances and several important known results are particular cases of more general results that are valid in this general framework.

The approximation property and Lipschitz mappings on Banach spaces

“…This notion was introduced by Carl and Stephani in [5] and the related approximation property has been studied in [10,13,14]. All these ideas have been unified in [4] where the concept of ideal topology has been introduced. Given I, J two ideals of linear operators, the unifying approximation property given in [4] reads as follows: a Banach space E is said to have the (I, J , τ )-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the ideal topology τ , by operators belonging to the operator ideal J .…”

“…All these ideas have been unified in [4] where the concept of ideal topology has been introduced. Given I, J two ideals of linear operators, the unifying approximation property given in [4] reads as follows: a Banach space E is said to have the (I, J , τ )-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the ideal topology τ , by operators belonging to the operator ideal J . Let us recall some concepts related to the tandem approximation property/operator ideals.…”

“…It is known that a linear map T : F → E between Banach spaces is I-Lipschitz compact if, and only if, it is linear I-compact. 4 The I-approximation property for Lipschitz operators.…”

Ideals of polynomials between Banach spaces revisited

“…The natural extension to multilinear operators and polynomials was designed by Pietsch some years later in [25]. Nowadays, ideals of polynomials and multilinear operators are explored by several authors in different directions (see, for instance, [1,5,6,8,9,10,11,12,13,14,20]). In this paper we are mainly interested in the theory of ideals of polynomials and ideals of multilinear operators between Banach spaces.…”

Adiposidade e história familiar de diabetes tipo 2 em uma população de adolescentes miscigenados

“…A estabilidade de um ideal de operadores pela formação do produto tensorial projetivo e pelo produto tensorial simétrico projetivo (as definições precisas serão dadas nesta seção) é uma ferramenta muito útil no estudo de classes de polinômios homogêneos (veja, por exemplo, [8,9,18] e as referências ali citadas). Usaremos nesta seção a estabilidade tensorial para obter diversos resultados sobre os germes de ideais gerados pelos adjuntos generalizados.…”