Quasi \(s\)-numbers and measures of non-compactness of multilinear operators (Q2860900)

Let \(X_1, \dots, X_m, Y\) be Banach spaces and \({\mathcal L}_m(X_1\times \dots\times X_m, Y)\) the Banach space of all \(m\)-linear bounded operators from \(X_1\times \dots\times X_m\) to \(Y\). Following the theory of \(s\)-numbers for linear operators, the authors introduce the notion of quasi \(s\)-numbers for multilinear operators. A mapping \(s=(s_n): {\mathcal L}_m(X_1\times \dots\times X_m, Y) \to [0,\infty)^{\mathbb N}\) which assigns to any operator \(T\in {\mathcal L}_m(X_1\times \dots\times X_m, Y)\) a sequence \((s_n(T))_{n\geq 1}\) of non-negative numbers is said to be an \(m\)-quasi \(s\)-number sequence if it satisfies the following conditions:{\parindent=8mm\begin{itemize}\item[(S1)] Monotonicity: \(\| T\|=s_1(T)\geq s_2(T)\geq \dots \geq 0\). \item[(S2)] Additivity: \(s_{k+n-1}(S+T)\leq s_k(S)+s_n(T)\). \item[(S3)] Ideal-property: \(s_n(ST)\leq \|S\| s_n(T)\).\item[(S4)] Rank-property: \(s_n(T)=0\) if \(\text{rank}(T)<n\).NEWLINENEWLINE\end{itemize}} If an \(m\)-quasi \(s\)-number sequence also satisfies the {\parindent=8mm\begin{itemize}\item[(S5)] Norming property: \(s_n(I)=1\), where \(I\) is the identity operator on the \(n\)-dimensional Hilbert space,NEWLINENEWLINE\end{itemize}}then it is called an \(s\)-number sequence.NEWLINENEWLINEIn this paper, the theory of quasi \(s\)-number sequences of bounded multilinear operators between Banach spaces is developed. The relationships among multilinear variants of approximation, Kolmogorov and Gelfand numbers of operators and their generalized linear adjoint are shown. Several results which are well-known in the linear case are stated and proved in the multilinear setting. Estimates of measures of non-compactness of multilinear operators in terms of measures of the adjoint operators are also given.