Unitary automorphisms of the space of $(T+H)$-matrices

Let $TH_n$ be the space of $(T+H)$-matrices of order $n$. The paper considers the following question: Which unitary matrices $U$ satisfy the condition $\forall A\in TH_n\to U^*AU\in TH_n$? A criterion for verifying whether a given matrix $U$ has this property is proposed.