Journal of Operator Theory
Volume 79, Issue 1, Winter 2018 pp. 55-77.
Some properties of the spherical $m$-isometries
Authors:
Karim Hedayatian (1) and Amir Mohammadi-Moghaddam (2)
Author institution:(1) Department of Mathematics, College of Sciences,
Shiraz University, Shiraz, 7146713565, IRAN
(2) Department of Mathematics, College of Sciences,
Shiraz University, Shiraz, 7146713565, IRAN
Summary: A commuting $d$-tuple $T=(T_{1}, \ldots, T_{d})$
is called a spherical $m$-isometry if
$\sum\limits_{j=0}^{m}(-1)^{j}\binom{m}{j}Q_{T}^{j}(I)=0$, where
$Q_{T}(A)=\sum\limits_{i=1}^{d}\!T_{i}^{*}AT_{i}$ for every
bounded linear operator $A$ on a Hilbert space $\mathcal{H}$.
Under some assumptions we prove that every power of $T$ is a spherical $m$-isometry.
Also, we study the products of spherical $m$-isometries when they remain spherical
$n$-isometries, for a suitable $n$. Besides,
we prove that the spherical $m$-isometries are power regular and for every proper
spherical $m$-isometry there are linearly independent operators
$A_{0},\ldots, A_{m-1}$ such that
$Q_{T}^{n}(I)=\sum\limits _{i=0}^{m-1}A_{i}n^{i}$ for every $n\geqslant 0$.
DOI: http://dx.doi.org/10.7900/jot.2016oct31.2149
Keywords: $m$-isometry, power regularity, spherical $m$-isometry, $d$-tuple
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