×

Powers of \(m\)-isometries. (English) Zbl 1256.47023

According to F. Bayart [Math. Nachr. 284, No. 17–18, 2141–2147 (2011; Zbl 1230.47018)], a continuous linear operator \(T\) on a Banach space \(X\) is an \((m,p)\)-isometry (\(m \in \mathbb{N}\), \(p>0\)) if, for each \(x \in X\), \(\sum_{k=0}^m \binom{m}{k} ||T^k x||^p = 0\). Continuing the work of P. Hoffmann, M. Mackey and M. Ó Searcóid [Integral Equations Oper. Theory 71, No. 3, 389-405 (2011; Zbl 1256.47002)] and of S. M. Patel [Glas. Mat., III. Ser. 37, No. 1, 141–145 (2002; Zbl 1052.47010)], the authors use the technique of recursive equations, that had already been used by Müller to study \(m\)-contractions, to prove the following two results. (1) If \(T\) is an \((m,p)\)-isometry, then every power \(T^s\) is also an \((m,p)\)-isometry. (2) If \(T^r\) is an \((m,p)\)-isometry and \(T^s\) is an \((l,p)\)-isometry, then \(T^t\) is an \((h,p)\)-isometry, with \(t\) the greatest common divisor of \(r\) and \(s\), and \(h\) the minimum of \(m\) and \(l\). In particular, if \(T^r\) and \(T^{r+1}\) are \((m,p)\)-isometries, then so is \(T\).

MSC:

47B99 Special classes of linear operators
PDFBibTeX XMLCite
Full Text: DOI