Compact operators in TRO's (Q2845743)

A ternary ring of operators (TRO, in short) between Hilbert spaces \(H_2\) and \(H_1\) is a norm closed subspace \(\mathcal V\) of \(B(H_2,H_1)\) so that \(xy^*z\in \mathcal V\) for every \(x,y,z\in\mathcal V\). Given a normed space \(X\), for a subset \(S\subset B_X\) the set of contractive perturbations of \(S\) is defined as NEWLINE\[NEWLINE cp(S)=\{x\in X:\|x\pm s\|\leq1\text{ for all }s\in S\}. NEWLINE\]NEWLINE Among other things, in this paper it is shown that, given a TRO \(\mathcal V\) and an element \(a\in \mathcal V\) with norm no greater than 1, the set of second order contractive perturbations \(cp^2(a)\) is weakly compact if and only if there exists a faithful representation \(\pi\) of \(\mathcal V\) such that \(\pi(a)\) is a compact operator. This is also equivalent to the fact that the operator \(x\mapsto ax^*a\) is compact in \(\mathcal V\).