On the representation of linear functionals on hyper-ideals of multilinear operators
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Tusi Mathematical Research Group
ORIGINAL PAPER
On the representation of linear functionals on hyper‑ideals of multilinear operators Geraldo Botelho1 · Raquel Wood2 Received: 2 September 2020 / Accepted: 22 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract A standard technique in infinite dimensional holomorphy, which produced several useful results, uses the Borel transform to represent linear functionals on certain spaces of multilinear operators between Banach spaces as multilinear operators. In this paper, we develop a technique to represent linear functionals, as linear operators, on spaces of multilinear operators that are beyond the scope of the standard technique. Concrete applications to some well-studied classes of multilinear operators, including the class of compact multilinear operators, and to one new class are provided. We can see, in particular, that sometimes our representations hold under conditions less restrictive than those of the related classical ones. Keywords Operator ideals · Multilinear operators · Borel transform · Hyper-ideals Mathematics Subject Classification 47H60 · 47A67 · 46G25 · 47L22 · 47B10
1 Introduction The representation of linear functionals is a classical and very useful topic in Functional Analysis and Operator Theory, going back to the celebrated representation theorems due to F. Riesz. Starting with the representation of functionals on sequence spaces and function spaces we all learned at graduate school, the subject spread through all subareas of mathematical analysis. In Infinite Dimensional Holomorphy, the representation of linear functionals on spaces of multilinear Communicated by Juan Seoane Sepúlveda. * Geraldo Botelho [email protected] Raquel Wood [email protected] 1
Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia 38.400‑902, Brazil
2
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo 05.508‑090, Brazil
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G. Botelho and R. Wood
operators, polynomials and holomorphic functions has been a valuable tool since the 1960s with the seminal works of Gupta [21, 22], who introduced the use of the Borel transform in the area. Several results followed, for example Dwyer’s and Carando–Dimant’s representations of functionals on spaces of nuclear multilinear operators/polynomials [8, 14], Alencar’s representation of linear functionals on spaces of multilinear operators/polynomials that can be approximated by finite type operators [1] and Matos’ representation of functionals on spaces of (s; r1 , … , rn )-nuclear multilinear operators [24]. The books [12, 13] by Dineen are excellent sources for results of this kind and their many applications. Recent developments, including applications to linear dynamics, can be found, e.g., in Refs. [2, 7, 15–18, 27]. Let us give a brief description of the Borel transform technique. By E∗ , we denote the (topological) dual of the Banach space E. Given a subspace M(E1 , … , En ; F) of the space of n-linear operators from E1 × ⋯ × En to F, wher
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