Products of \(m\)-isometries
Let \(X\) be a Banach space, \(m \in \mathbb N\) and \(1 \leq p < \infty\). A bounded linear operator \(T \in L(X)\) is said to be an \((m,p)\)-isometry if \[ \sum_{k=0}^m \binom{m}{k} (-1)^{m-k} \| T^kx\|^p = 0 \] for all \(x \in X\). For a Hilbert space \(X\) and \(p = 2\), one obtains the class of \(m\)-isometries introduced by \textit{J. Agler} [Am. J. Math. 112, No. 1, 1--14 (1990; Zbl 0708.47020)]. The authors show here that, if \(T \in L(X)\) is an \((n,p)\)-isometry, \(S \in L(X)\) is an \((m,p)\)-isometry and \(ST = TS\), then \(ST\) is an \((n+m-1,p)\)-isometry. As an application, it follows that, if \(H\) is a Hilbert space, \(T \in L(H)\) is an \(n\)-isometry and \(S^* \in L(H)\) is an \(m\)-isometry, then the two-sided multiplication operator \(A \mapsto TAS\) is an \((n+m-1)\)-isometry \(C_2(H) \to C_2(H)\), where \(C_2(H)\) is the space of Hilbert-Schmidt operators on \(H\).
scientific article; zbMATH DE number 6127325
scientific article