Scalar-valued dominated polynomials on Banach spaces

Abstract

Abstract. It is well known that 2-homogeneous polynomials on L ∞ -spaces are 2-dominated. Motivated by the fact that related coincidence results are possible only for polynomials defined on symmetrically regular spaces, we investigate the situation in several classes of symmetrically regular spaces. We prove a number of non-coincidence results which makes us suspect that there is no infinite dimensional Banach space E such that every scalar-valued homogeneous polynomial on E is r-dominated for every r ≥ 1.

Dominated Bilinear Forms and 2-homogeneous Polynomials

“…As usual we write P d,r ( m X) and P( m X) when Y = K. The definition (and notation) for r-dominated multilinear mappings is analogous (for the notation just replace P by L). For details we refer to [2,4,11].…”

“…So, coincidence situations can occur only for m = 2. Sometimes it happens that every continuous bilinear form on X 2 is 2-dominated, for example if X is either an L ∞ -space, the disc algebra A or the Hardy space H ∞ (see [4,Proposition 2.1]). In this case every continuous bilinear form on X 2 and every continuous scalarvalued 2-homogeneous polynomial on X are r-dominated for every r ≥ 2.…”

Summability of multilinear mappings: Littlewood, Orlicz and beyond

“…These two important results can be considered as the roots of what has been known as coincidence and non-coincidence results. The passage from the linear to the multilinear case has occasioned the emergence of several coincidence and non-coincidence situations for absolutely summing multilinear mappings (see [1,7,11,12,13,30,31,32,33,34]). The scope of the present paper is to prove new coincidence theorems, some of them generalizing known results and some giving new perspectives to the subject.…”

Absolutely summing linear operators into spaces with no finite cotype

“…Continuous bilinear forms on either an L ∞ -space, or the disc algebra A or the Hardy space H ∞ are 2-dominated [4, Proposition 2.1]. On the other hand, partially solving a problem posed in [4], in [10, Lemma 5.4] it was recently shown that for every n ≥ 3, every infinite-dimensional Banach space X and any p ≥ 1, there is a continuous n-linear form on X n which fails to be p-dominated. As to vector-valued bilinear mappings, all that is known, as far as we know, is that for every L ∞ -spaces X 1 , X 2 , every infinite-dimensional space Y and any p ≥ 1, there is a continuous bilinear mapping A : X 1 × X 2 → Y which fails to be p-dominated [3,Theorem 3.5].…”