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 $P\colon\mathcal{A}(X)\to Y$ 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 $T\in\mathcal{A}(X)$.