Multiple splitting property for dominated bilinear operators

Abstract

We prove that if $1\le p<q,r<\infty$ are such that $\frac{1}{p}=\frac{1}{q} +\frac{1}{r}$ then, each $\left( r,p\right)$-dominated (or $\left( p,r\right)$-dominated) operator $U:X\times Y\rightarrow Z$ has the multiple splitting property: for every $\left( x_{i}\right) _{1\le i\le n}\subset X$, $\left( y_{j}\right) _{1\le j\le m}\subset Y$, there exists $\left( \lambda _{i}\right) _{1\le i\le n}\subset {\mathbb {K}}$, $\left( \nu _{j}\right) _{1\le j\le m}\subset {\mathbb {K}}$ and $\left( z_{ij}\right) _{1\le i\le n;1\le j\le m}\subset Z$ such that

$$\begin{aligned} U\left( x_{i},y_{j}\right) =\lambda _{i}\nu _{j}z_{ij}\text { for }1\le i\le n,1\le j\le m;\text {\ }l_{r}\left( \lambda _{i}\right) \le 1, \, l_{r}\left( \nu _{j}\right) \le 1;\, \end{aligned}$$

and

$$\begin{aligned} w_{q}\left( z_{ij}\mid 1\le i\le n;1\le j\le m\right) \le Cw_{p}\left( x_{i}\mid 1\le i\le n\right) w_{p}\left( y_{j}\mid 1\le j\le m\right) \end{aligned}$$

where $C=\Delta _{r,p}\left( U\right)$ (or $\Delta _{p,r}\left( U\right)$ ). As a corollary we deduce the well known Maurey–Pietsch splitting property in the linear case. As applications we prove new Pietsch’s composition type results for multiple summing operators which have no linear analogue. Further we show that the classical Pietsch composition result and other new composition results for multiple summing operators can be deduced from our multiple splitting property.