Invertible weighted shift operators which are \(m\)-isometries (Q2855903)

Let \(H\) be a complex Hilbert space. For a bounded linear operator \(T\) on \(H\) and a positive integer \(m\), one defines NEWLINE\[NEWLINE \Delta_{T,m}=\sum_{k=1}^{m}(-1)^k{m \choose k} T^{*\, m-k}T^{m-k}. NEWLINE\]NEWLINE If \(\Delta_{T,m}=0\), then \(T\) is said to be an \(m\)-isometry. The following is proved:NEWLINENEWLINE(i) for any even positive integer \(m\), there exists an invertible \((m+1)\)-isometry which is not an \(m\)-isometry;NEWLINENEWLINE(ii) a power bounded \(m\)-isometry is an isometry;NEWLINENEWLINE(iii) if \(m\geq 2\) and \(T\) is an \(m\)-isometry, then \(\Delta_{T,m-1}\) is not invertible;NEWLINENEWLINE(iv) if \(T\) is an \(m\)-isometry, then \(T^n\) is an \(m\)-isometry for every positive integer \(n\).