Abstract
Let S be a n-by-n truncated shift whose numerical radius equal one. First, Cassier et al. (J Oper Theory 80(2):453–480, 2018) proved that the Harnack part of S is trivial if \(n=2\), while if \(n=3\), then it is an orbit associated with the action of a group of unitary diagonal matrices; see Theorem 3.1 and Theorem 3.3 in the same paper. Second, Cassier and Benharrat (Linear Multilinear Algebra 70(5):974–992, 2022) described elements of the Harnack part of the truncated n-by-n shift S under an extra assumption. In Sect. 2, we present useful results in the general finite-dimensional situation. In Sect. 3, we give a complete description of the Harnack part of S for \(n=4\), the answer is surprising and instructive. It shows that even when the dimension is an even number, the Harnack part is bigger than conjectured in Question 2 and we also give a negative answer to Question 1 (the two questions are contained in the last cited paper), when \(\rho =2\).
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20 February 2024
A Correction to this paper has been published: https://doi.org/10.1007/s43036-024-00322-z
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Acknowledgements
The second and third authors gratefully acknowledge the financial support from the Laboratory of Fundamental and Applicable Mathematics of Oran (LMFAO) and the Algerian research project: PRFU, no. C00L03ES310120220003 (D.G.R.S.D.T).
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Communicated by Martin Mathieu.
The original online version of this article was revised: On page 12 of the article, in the matrix of the third line counting from the bottom should put “\(\overline{z} B \textbf{R}\)” instead of “\(B \textbf{R}\)”.
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Cassier, G., Naimi, M. & Benharrat, M. Harnack parts for 4-by-4 truncated shift. Adv. Oper. Theory 9, 11 (2024). https://doi.org/10.1007/s43036-023-00309-2
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DOI: https://doi.org/10.1007/s43036-023-00309-2