On the decomposition of families of quasinormal operators (Q2851288)

The paper under review discusses several decomposition theorems for a commuting family of quasinormal operators. Recall that \(T\) is \textit{quasinormal} if \([T^*, T]T=0\), where \([T^*, T]\) denotes the commutator \(T^*T-TT^*\) of \(T\). There is one central decomposition theorem to be referred to as multiplie injective canonical decomposition theorem: Let \(\{T_i\}_{i \in Z}\) be a family of jointly quasinormal operators on a separable Hilbert space \(H\), where \(Z\) is a finite or infinite subset of \(\mathbb Z\). Then there is a decomposition NEWLINE\[NEWLINEH =\bigoplus_{\alpha \in \{0, 1\}^Z}H_{\alpha}NEWLINE\]NEWLINE into subspaces \(H_{\alpha}\) reducing the family \(\{T_i\}_{i \in Z}\), where \(T_i|_{H_{\alpha_i}}=0\) for \(\alpha_i=0\) and \(\alpha^{-1}_iT_i|_{H_{\alpha_i}}\) is injective for \(\alpha_i=1\). The author also provides a generalization of this decomposition to a pair of commuting quasinormal partial isometries.