Orthogonally additive polynomials on the algebras of approximable operators

TL;DR

This paper characterizes orthogonally additive polynomials on the algebra of approximable operators, showing they are uniquely determined by a linear map when the dual space has the bounded approximation property.

Contribution

It proves that such polynomials are represented by a linear map composed with the nth power, under the bounded approximation property condition.

Findings

Existence of a unique linear map representing the polynomial.

Representation holds for all operators in the algebra.

Requires the dual space to have the bounded approximation property.

Abstract

Let $X$ and $Y$ be Banach spaces, let $\mathcal{A}(X)$ stands for the algebra of approximable operators on $X$, and let $P\colon\mathcal{A}(X)\to Y$ be an orthogonally additive, continuous $n$-homogeneous polynomial. If $X^*$ has the bounded approximation property, then we show that there exists a unique continuous linear map $\Phi\colon\mathcal{A}(X)\to Y$ such that $P(T)=\Phi(T^n)$ for each $T\in\mathcal{A}(X)$.