A Wold-type decomposition for a class of \(\nu\)-hypercontractions (Q2835232)

The content of this paper in multi-dimensional operator theory is best described by its abstract:NEWLINENEWLINE``For a positive integer \(k\) and a \(d\)-tuple \(T=(T_1,\dots,T_d)\), consider NEWLINE\[NEWLINED_{T,k}:=\sum_{l=0}^k (-1)^l {k\choose l}\sum_{| p| =l} \frac{l!}{p!} {T^*}^pT^p.NEWLINE\]NEWLINE A commuting \(d\)-tuple \(T\) is said to be a row \(\nu\)-hypercontraction if \(D_{T^*,k}\geq 0\) for \(k=1,\dots, \nu\). Under some assumption, we prove that any row \(\nu\)-hypercontraction \(d\)-tuple \(T\) for which \(D_{T^*,\nu}\) is a projection, decomposes into \(S_\nu\oplus V^\ast\) for a direct sum \(S_\nu\) of a `\(d\)-multiplication tuple' \(M_{z,\nu}\) and a spherical isometry \(V\). In addition, if \(T\) is a spherical expansion and \(d\geq \nu\), then \(T=S_\nu\oplus U\) for a spherical unitray \(U\). This generalizes a theorem of \textit{S. Richter} and \textit{C. Sundberg} [J. Funct. Anal. 258, No. 10, 3319--3346 (2010; Zbl 1195.47006)]. Further, we identify extremals of joint \(\nu\)-hypercontractive \(d\)-tuples.