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Some techniques on nonlinear analysis and applications. (English) Zbl 1248.47024
The main aim of this interesting article is to present in a context of strong generality (that is, arbitrary mappings between spaces with almost no structure) versions of two classical results about linear operators between Banach spaces.
The first one is the Inclusion Theorem for absolutely summing mappings, originally proved by S. Kwapień in [“Some remarks on \((p,q)\)-absolutely summing operators in \(\ell_{p}\)-spaces”, Stud. Math. 29, 327–337 (1968; Zbl 0182.17001)]. It states the following: for \(1\leq p_1\leq q_1<\infty\), \(1\leq p_2\leq q_2<\infty\), if a continuous linear operator between Banach spaces \(u:X\to Y\) satisfies that there exists \(C>0\) such that \[ \left(\sum_{j=1}^m \|u(x_j)\|^{q_1}\right)^{1/q_1}\leq C \sup_{\varphi\in B_{X^*}}\left(\sum_{j=1}^m |\varphi(x_j)|^{p_1}\right)^{1/p_1} \] for every \(m\) and all \(x_1,\dots, x_m\in X\), then it should also satisfy \[ \left(\sum_{j=1}^m \|u(x_j)\|^{q_2}\right)^{1/q_2}\leq C \sup_{\varphi\in B_{X^*}}\left(\sum_{j=1}^m |\varphi(x_j)|^{p_2}\right)^{1/p_2} \] for every \(m\) and all \(x_1,\dots, x_m\in X\), provided that \[ p_1\leq p_2,\qquad q_1\leq q_2\qquad\text{ and }\qquad \frac{1}{p_1}-\frac{1}{q_1}<\frac{1}{p_2}-\frac{1}{q_2}. \] Its direct proof, which combines Hölder’s inequality with the homogeneity of linear operators, can be adapted for multilinear maps or homogeneous polynomials. The authors here formulate a slightly weaker statement for arbitrary mappings between general spaces and derive from it the stronger multilinear result. Naturally, this broad presentation could also be useful to obtain potential extensions to other unexplored settings.
Before illustrating some of the applications, the authors present an enjoyable and careful review of the emergence and development of the concept of absolute summability for linear operators and of some of its multilinear versions.
The second result exposed here under broad conditions is the classical Pietsch Domination Theorem for absolutely summing linear operators [A. Pietsch, “Absolut \(p\)-summierende Abbildungen in normierten Räumen”, Stud. Math. 28, 333–353 (1967; Zbl 0156.37903)]. In the last decades, several extensions and variations of the concept of absolute summability have been developed for different classes of non-linear mappings and in almost every of such contexts a kind of Pietsch Domination Theorem was proved. All these results are particular cases of the “full general Pietsch Domination Theorem” presented in this paper.

MSC:
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47L22 Ideals of polynomials and of multilinear mappings in operator theory
47H99 Nonlinear operators and their properties
47H60 Multilinear and polynomial operators
46G25 (Spaces of) multilinear mappings, polynomials
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