Unitary automorphisms of the subspace of (T+H)-matrices of order 4
Consider matrices $U$ in the unitary group ${\bf U}_4$ for which $\forall A \in {\cal TH}_4 \longrightarrow B = U^*AU \in {\cal TH}_4$. Here, ${\cal TH}_4$ is the set of (T+H)-matrices of order four. Such matrices can be identified with unitary automorphisms of the subspace ${\cal TH}_4$. The following problem is treated: Can the boundary of $U$ not contain any zeros? (The boundary of a matrix is the collection of its entries standing in the first and last rows and the first and last columns.) We show that matrices $U$ with an entirely nonzero boundary do exist, which contrasts with the situation for the unitary automorphisms of the subspaces of Toeplitz and Hankel matrices.