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 $\mathrm{\Phi }:\mathcal{A}(X)\to Y$ such that $P(T)=\mathrm{\Phi }(T^n)$ for each $T\in \mathcal{A}(X)$.

Keywords:

• Algebra of approximable operators
• bounded approximation property
• orthogonally additive polynomial

AMS Subject Classifications:

• 47H60
• 46H35
• 47L10

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first and the third named authors were supported by MINECO [grant number MTM2015–65020–P]; Junta de Andalucía [grant number FQM–185]. The second named author was supported by Beca de iniciación a la investigación of Universidad de Granada.