| Abstrakt: |
Let Sbe a n-by-ntruncated shift whose numerical radius equal one. First, Cassier et al. (J Oper Theory 80(2):453–480, 2018) proved that the Harnack part of Sis 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-nshift Sunder 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 Sfor 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 ρ=2. |
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