Quasisimilarity of invariant subspaces for uniform Jordan operators of infinite multiplicity (Q1923693)

Let \(T\) be a bounded linear operator on a separable Hilbert space \({\mathcal H}\), and let \({\mathcal M}\) and \({\mathcal N}\) be two invariant subspaces for \(T\). The authors call \({\mathcal M}\) and \({\mathcal N}\) quasisimilar if there exist quasiaffinities (i.e. bounded one-to-one operators with dense ranges) \(X\) and \(Y\), commuting with \(T\), such that \((X,{\mathcal M})^-={\mathcal N}\) and \((Y,{\mathcal N})^-={\mathcal M}\). Recall that operators \(T\) and \(T'\) are quasisimilar if there exist quasiaffinities \(X:{\mathcal H}\to{\mathcal H}'\) and \(Y:{\mathcal H}'\to{\mathcal H}\) such that \(XT=T'X\) and \(YT'=TY\). They show that if \(T\) is a uniform Jordan operator of infinite multiplicity, then \({\mathcal M}\) is quasisimilar to \({\mathcal N}\) if and only if \(T|{\mathcal M}\) is quasisimilar to \(T|{\mathcal N}\) and \((T^*|{\mathcal M}^\perp)^*\) is quasisimilar to \((T^*|{\mathcal N}^\perp)^*\).