Powers of \(A\)-\(m\)-isometric operators and their supercyclicity (Q310690)

Let \(m\geq 1\) be an integer, and let \(A\in\mathcal{B}(H)^{+}\) be a positive bounded operator on a complex Hilbert space \(H\). An operator \(T\in\mathcal{B}(H)\) is called an \(A\)-\(m\)-isometry if \[ \sum_{k=0}^{m}\binom{m}{k}T^{*m-k}AT^{m-k}=0. \] This notion generalizes that of an \(m\)-isometry. The author studies products of \(A\)-\(m\)-isometries, and shows that powers of \(A\)-\(m\)-isometries are \(A\)-\(m\)-isometries. It is also proved that \(A\)-\(m\)-isometries are never supercyclic.