Orthogonally Additive Mappings on Hilbert Modules

TL;DR

This paper characterizes continuous orthogonally additive mappings on Hilbert modules over certain operator algebras, showing they can be decomposed into additive and linear parts.

Contribution

It provides a new representation theorem for orthogonally additive mappings on Hilbert modules over $ ext{K}( ext{H})$ and $ ext{S}( ext{H})$, detailing their structure.

Findings

Every continuous orthogonally additive mapping has a specific decomposition.

The mappings can be expressed as a sum of a continuous additive and a continuous linear map.

The result applies to modules over $ ext{K}( ext{H})$ and $ ext{S}( ext{H})$.

Abstract

In this paper, we study the representation of orthogonally additive mappings acting on Hilbert $C^*$-modules and Hilbert $H^*$-modules. One of our main results shows that every continuous orthogonally additive mapping $f$ from a Hilbert module $W$ over $\K(\H)$ or $\H\S(\H)$ to a complex normed space is of the form $f(x)=T(x)+\Phi(< x, x >)$ for all $x\in W$, where $T$ is a continuous additive mapping, and $\Phi$ is a continuous linear mapping.