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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