Harnack parts for 4-by-4 truncated shift

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 \(\rho =2\).