Limit Orders and Multilinear Forms on $\ell_p$ Spaces

Abstract

Since the concept of limit order is a useful tool to study operator ideals, we propose an analogous definition for ideals of multilinear forms. From the limit orders of some special ideals (of nuclear, integral, r-dominated and extendible multilinear forms) we derive some properties of them and show differences between the bilinear and n-linear cases (n ≥ 3).

Ideals of Multilinear Forms – a Limit Order Approach

“…With this we obtain the r-dominated multilinear mappings. In [4] the limit order of r-dominated n-linear forms was computed. Our aim in this section is to relate the adjoint ideal of r-dominated forms with the ideal of r-integral forms (see definitions below) and use the results in [4] and in the previous sections to compute the limit order of the latter.…”

“…In [4], the limit order for ideals of multilinear forms is defined. We recall the definitions and some basic facts.…”

“…Following the spirit of the definition of limit order for operator ideals in [4], the concept of limit order was defined for ideals of multilinear forms. First of all, if T ∈ L( n p ), we call it diagonal if there exists a sequence α = (α(k)) k such that for all x 1 , .…”

“…Examples: In [4] the limit orders of nuclear, integral and r-dominated multilinear forms (see [8] for definitions) are computed:…”

“…The concept of limit order for multilinear forms was introduced in [4]. As an application, it could be shown that some properties of bilinear ideals were no longer valid for the n-linear case (n ≥ 3).…”