Centrosymmetric property of unitary matrices that preserve the set of $(T+H)$-matrices under similarity transformations

The following problem is discussed: what are unitary $n\times n$ matrices $U$ that map the linear space of $(T+H)$-matrices into itself by similarity transformations? Analogous problems for the spaces of Toeplitz and Hankel matrices were solved recently. For $(T+H)$-matrices, the problem of describing appropriate matrices $U$ appears to be considerably more complex and is still open. The result proved in this paper may contribute to the complete solution of this problem. Namely, every such matrix $U$ is either centrosymmetric or skew-centrosymmetric; moreover, only the first variant is possible for odd $n$.