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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