On properties of linear operators of certain classes in rigged spaces with indefinite metric (Q1187748)

A linear space \({\mathcal N}={\mathcal N}_ ++{\mathcal N}_ -\) (algebraic direct sum) is called \({\mathcal J}_ \nu\)-space (with positive integer \(\nu\)) if there is a norm \(\|\cdot\|\) on \({\mathcal N}\) such that \(\| x_ ++x_ -\|^ 2=\| x_ +\|^ 2 +\| x_ -\|^ 2\) for \(x_ \pm\in{\mathcal N}_{\pm}\) and \({\mathcal N}\) is equipped with the functional \[ {\mathcal J}_ \nu(x_ +x_ -)\equiv\| x_ +\|^ \nu -\| x_ -\|^ \nu \quad \text{ for } \quad x_ \pm\in{\mathcal N}_ \pm. \] A \({\mathcal J}_ \nu\)-space \({\mathcal N}\) is called a rigged space if there is a Banach space norm \(\|\cdot \|_ 1\) on \({\mathcal N}\) such that \(\| x\|_ 1\geq \| x\|\) for \(x\in{\mathcal N}\). A linear operator \(V\) on a \({\mathcal J}_ \nu\)-space is called \({\mathcal J}_ \nu\)-semiunitary if \(V\) is defined on the whole space and \({\mathcal J}_ \nu\)-preserving. A \({\mathcal J}_ \nu\)-semiunitary operator needs not be \(\|\cdot\|\)- bounded. The authors present several sufficient conditions for boundedness of a \({\mathcal J}_ \nu\)-semiunitary operator on a rigged space and study implication relations among those conditions.