Orthogonally additive polynomials on the algebras of approximable operators

Abstract

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

To appear in Linear and Multilinear Algebra