# 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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