Positive multiple summing and concave multilinear operators on Banach lattices (Q2018694)

The authors first introduce the class \(\Lambda _{p}^{\text{mult}}( E_{1},\dots{},E_{n};Y) \) of positive multiple \(p\)-summing \(n\)-linear operators and the class \({C}_{p}^{\text{mult}}( E_{1},\dots{},E_{n};Y) \) of multiple \(p\)-concave \(n\)-linear operators. Next, they establish some inclusion results between the class \(\Pi _{p}^{\text{mult}}( E_{1},\dots{},E_{n};Y) \) of multiple \(p\)-summing \(n\)-linear operators and the two concepts cited above. If \(1\leq p<\infty \), they prove in the main theorem of Section 3 that \[ {C}_{1}^{\text{mult}}( E_{1},\dots{},E_{n};Y) \subset {C} _{p}^{\text{mult}}( E_{1},\dots{},E_{n};Y) . \] The same result is verified for positive multiple \(p\)-summing \(n\)-linear operators. If \(1\leq p<q<2\), then \({C}_{p}^{\text{mult}}( E_{1},\dots{},E_{n};Y) \subset {C}_{q}^{\text{mult}}( E_{1},\dots{},E_{n};Y)\); if \(Y\) has cotype \(2\), then \({C}_{p}^{\text{mult}}( E_{1},\dots{},E_{n};Y) \subset {C} _{2}^{\text{mult}}( E_{1},\dots{},E_{n};Y)\) for \(1\leq p<\infty \). Another result announces that, if \(E_{1},\dots{},E_{n}\) are AL-spaces, we have for \(1\leq p<\infty \) \[ \Lambda _{p}^{\text{mult}}( E_{1},\dots{},E_{n};Y) ={C} _{p}^{\text{mult}}( E_{1},\dots{},E_{n};Y) ={L}( E_{1},\dots{},E_{n};Y). \] The last part of this paper is devoted to study the class \(\Lambda _{1}^{\text{mult}}( E_{1},\dots{},E_{n};F) \). If \(F\) is a dual Banach lattice, then \( \Lambda _{1}^{\text{mult}}( E_{1},\dots{},E_{n};F) \) is also dual Banach lattice and an ideal of \({L}^{r}( E_{1},\dots{},E_{n};F) \), the space of all regular \(n\)-linear operators. If, moreover, \(F\) is an AL-space, then \( \Lambda _{1}^{\text{mult}}( E_{1},\dots{},E_{n};F) \) coincides with \({L} ^{r}( E_{1},\dots{},E_{n};F)\).