[Paper Review] Polynomial ideals from a nonlinear viewpoint

Geraldo Botelho, Ewerton R. Torres|arXiv (Cornell University)|Mar 1, 2016

Advanced Banach Space Theory53 references3 citations

TL;DR

This paper introduces and systematically studies polynomial two-sided ideals—classes of homogeneous polynomials closed under composition with other homogeneous polynomials—extending the classical theory of polynomial ideals, which only require closure under composition with linear operators. The key contribution is establishing that tensorstable operator ideals generate polynomial two-sided ideals, with explicit examples including duals of absolutely summing and compact operators, and proving that s-tensorstability implies the desired closure property.

ABSTRACT

Classes of homogeneous polynomials between Banach spaces have been studied in the last three decades from the perspective of the so-called ideal property: if a polynomial P belongs to a class Q, then the composition u o P o v of P with linear operators u and v belongs to Q as well. In an attempt to explore the nonlinearity of the subject in a more consistent way, and taking into account recent results in the field, in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials. Some importante classes justify the study of the intermediate concept of classes of polynomials Q such that if P belongs to Q, u is a linear operator and Q is a homogeneous polynomial, then u o P o Q belongs to Q.

Motivation & Objective

• To address the linear bias in classical polynomial ideal theory by introducing a more nonlinear framework based on composition with homogeneous polynomials.
• To define and study polynomial two-sided ideals, where classes are closed under composition with arbitrary homogeneous polynomials.
• To introduce the intermediate concept of polynomial hyper-ideals, closed under composition with one linear and one homogeneous polynomial.
• To establish connections between multilinear hyper-ideals and polynomial hyper-/two-sided ideals.
• To provide a systematic construction of new and known classes as polynomial two-sided ideals via tensorstable operator ideals.

Proposed method

• Define polynomial two-sided ideals as classes of homogeneous polynomials closed under composition with arbitrary homogeneous polynomials.
• Introduce polynomial hyper-ideals as classes closed under composition with one linear and one homogeneous polynomial.
• Establish a link between multilinear hyper-ideals and polynomial hyper-ideals via the generation of polynomial classes from multilinear ones.
• Prove that if an operator ideal is tensorstable, then the associated polynomial class is s-tensorstable, which implies closure under composition with homogeneous polynomials.
• Use the symmetrization operator and inclusion maps to relate the projective symmetric tensor product to the full projective tensor product.
• Apply known results on tensorstability of operator ideals (e.g., duals of absolutely summing, nuclear, and integral operators) to generate new polynomial two-sided ideals.

Experimental results

Research questions

• RQ1 Which classes of homogeneous polynomials are closed under composition with arbitrary homogeneous polynomials, and how do they differ from classical polynomial ideals?
• RQ2 Can the theory of multilinear hyper-ideals be systematically extended to the setting of polynomial two-sided ideals?
• RQ3 What conditions on an operator ideal ensure that the associated polynomial class forms a two-sided ideal?
• RQ4 How do classical polynomial ideals like weakly compact or absolutely summing polynomials behave under the new two-sided ideal framework?
• RQ5 Is every s-tensorstable operator ideal necessarily tensorstable, and what are the implications for the construction of polynomial two-sided ideals?

Key findings

• The class of hyper-nuclear polynomials is the smallest polynomial two-sided ideal and also the smallest polynomial hyper-ideal.
• Every tensorstable operator ideal generates a Banach polynomial two-sided ideal via the composition construction $\mathcal{I} \circ \mathcal{P}$.
• The dual $\Pi_p^{\text{dual}}$ of the ideal of absolutely summing $p$-operators is tensorstable, hence generates a polynomial two-sided ideal.
• The ideal $\mathcal{S}$ of separable operators and the ideals $\overline{\mathcal{F}}^{\|\cdot\|}$, $\mathcal{N}$, $\mathcal{J}$, $\mathcal{L}_{\infty,q,\gamma}$, $\mathcal{L}_{1,q}$, $\mathcal{K}_{1,p}$, $\mathcal{J}_p^{\text{dual}}$, and $\mathcal{K}_{1,p}^{\text{sur}}$ are all tensorstable and thus generate polynomial two-sided ideals.
• The paper proves that every tensorstable operator ideal is s-tensorstable, and this implies that $\mathcal{I} \circ \mathcal{P}$ is a polynomial two-sided ideal, providing a general method for constructing such classes.