Journal of Operator Theory

Volume 75, Issue 1, Winter 2016 pp. 195-208.

A Wold-type decomposition for a class of row $\nu$-hypercontractions

Authors: Sameer Chavan (1) and Rani Kumari (2)
Author institution: (1) Indian Institute of Technology Kanpur, Kanpur- 208016, India
(2) Indian Institute of Technology Kanpur, Kanpur- 208016, India

Summary: For a positive integer $k$ and $d$-tuple $T=(T_1, \ldots, T_d)$, consider $ D_{T, k}:= \sum\limits_{l=0}^{k} (-1)^l {k \choose l} {\sum\limits_{|p|=l}\frac{l!}{p!}{T^*}^p}{T^p}. $ A commuting $d$-tuple $T$ is said to be a row $\nu$-hyper\-contraction if $D_{T^*, k} \geqslant 0$ for $k = 1, \ldots, \nu.$ Under some assumption, we prove that any row $\nu$-hypercontraction $d$-tuple $T$, for which $D_{T^*, \nu}$ is a projection, decomposes into $S_{\nu} \oplus V^*$ for a direct sum $S_{\nu}$ of $M_{z,\nu}$ and a spherical isometry $V$. In addition, if $T$ is a spherical expansion and $d \geqslant \nu,$ then $T= S_{\nu} \oplus U$ for a spherical unitary $U$. This generalizes a theorem of Richter-Sundberg. Further, we identify extremals of joint $\nu$-hypercontractive $d$-tuples.

DOI: http://dx.doi.org/10.7900/jot.2015jan17.2066
Keywords: Wold-type decomposition, hypercontraction, row $\nu$-hypercontraction, extremal family

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