Dominated Bilinear Forms and 2-homogeneous Polynomials

Abstract

The main goal of this note is to establish a connection between the cotype of the Banach space X and the parameters r for which every 2-homogeneous polynomial on X is rdominated. Let cot X be the infimum of the cotypes assumed by X and (cot X) * be its conjugate. The main result of this note asserts that if cot X > 2, then for every 1 ≤ r < (cot X)* there exists a non-r-dominated 2-homogeneous polynomial on X.2010 Mathematics Subject Classification: 46G25, 46B20, 46B28. Keywords: r-dominated multilinear form, r-dominated homogeneous polynomial, absolutely (p; q)-summing mapping, cotype. §1. IntroductionThe notion of p-dominated multilinear mappings and homogeneous polynomials between Banach spaces plays an important role in the nonlinear theory of absolutely summing operators. It was introduced by Pietsch [17] and has been investigated by several authors since then (see, e.g., [5,6] and references therein). Let X be a Banach space and m be a positive integer. A continuous m-are weakly r-summable in X. In a similar way, a scalarvalued m-homogeneous polynomial P on X is r-dominated if (P (x j )) ∞ j=1 ∈ r/m whenever (x j ) ∞ j=1 is weakly r-summable in X.

Factorization of absolutely continuous polynomials

“…It is clear that, so defined, v is continuous and closes the diagram (9). The converse follows from Proposition 4.3 and the ideal property.…”

“…On the other hand, in the last ten years a considerable effort has been made in order to increase the knowledge on the polynomials that belong to some operator ideal (see for instance [2,3,4,12,22] and the references therein). The case of the p-dominated polynomials is particularly relevant and has been intensively studied ( [6,7,8,9,10]). …”

Absolutely summing multilinear operators: a panorama

“…The existence of spaces fulfilling the hypotheses of Theorem 5.7 is assured by G. Pisier [83]. Also, cot E = 2 is a necessary condition for Theorem 5.7 since in [24] it is also proved that…”

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

“…We recall that P ∈ P( m X; Y ) is absolutely (r, s)summing (in symbols P ∈ P as(r,s) ( m X; Y )) if the sequence (P (x j )) ∞ j=1 belongs to ℓ r (Y ) whenever (x j ) ∞ j=1 is weakly s-summable in X (see [2]). In [12,Theorem 3.1] it is proved that if m is an even integer, if Z is an infinite dimensional real Banach space and if r < 1, then the coincidence P ( m Z; R) = P as(r,s) ( m Z; R) implies that Id Z is mr 1−r , ssumming. A careful examination of [12,Theorem 3.1] shows that the argument of [12] cannot be extended to the case of odd integers and complex scalars.…”