Hyper-ideals of multilinear operators

Abstract

We introduce and develop the notion of hyper-ideals of multilinear operators between Banach spaces. While the well studied notion of ideals of multilinear operators (multi-ideals) relies on the composition with linear operators, the notion we propose, by considering the composition with multilinear operators, explores more deeply the nonlinear feature of the subject. The results we prove show that, although more restrictive a priori, hyper-ideals enjoy nice general properties and its theory is rich enough to provide distinguished examples in every situation where the corresponding multi-ideal fails to be a hyper-ideal.

Coherence and compatibility: a stronger approach

“…The composition ideal is a hyper-ideal; for more details see [7]. In that work we also see that, when (I, · I ) is an p-Banach ideal, then the composition ideal is also an p-Banach hyper-ideal.…”

“…Our work is motivated in the interest of comparing distinct levels of linearity that are not neighbors and this comparison is made through forms (scalar polynomials) rather than the product of linear functionals (linear functional powers). One of the motivations for this approach is the concept of hyperideals, introduced in [7], where multilinear mappings classes satisfy a stronger property than the multi-ideal property. Instead of the class being stable by composition with linear operators, the class is stable by composition with multilinear mappings, and such a concept will be adapted to homogenous polynomials.…”

The duality between ideals of multilinear operators and tensor norms

“…As the reader knows, there exist other approaches of ideals of multilinear operators highly influenced by ideas of Pietsch [18]. This is the case of hyper-ideals developed by Botelho and Torres [4] and multiideals developed by Floret and Hunfeld [13]. This late is the most popular notion and many of known examples of classes of multilinear operators fit in this framework.…”

Representação de funcionais lineares em hiper-ideais de operadores multilineares e de polinômios homogêneos

“…Da mesma forma, as transformadas de Borel multilinear e polinomial funcionam bem para representar funcionais lineares em ideais de operadores multilineares (multiideais) e em ideais de polinômios, pois os multi-ideais contêm os operadores de tipo finito e os ideais de polinômios contêm os polinômios de tipo finito. Já os hiper-ideais de operadores multilineares e de polinômios homogêneos (veja [20,21,72]) contêm os operadores multilineares e os polinômios de posto finito, o que faz com que esses ideais sejam os objetos de aplicação das hiper-transformadas de Borel multilinear e polinomial. Descrevemos a seguir como estruturamos a tese para alcançar o objetivo proposto.…”

“…. , r n )nucleares, essa nova classe foi introduzida e estudada em [20,72] sob o nome de operadores hiper-(s; r)-nucleares. Provaremos no Capítulo 2 alguns resultados que prepararão essa classe para a aplicação da hiper-transformada de Borel.…”