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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=(T1,,Td) 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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