On commutative algebras of Toeplitz-plus-Hankel matrices

It is known that the entire class of Hermitian Toeplitz matrices can be mapped into a subset of real Toeplitz-plus-Hankel matrices ($(T + H)$-matrices) by one and the same unitary similarity transformation. This fact is refined by showing that the resulting $(T + H)$-matrices are symmetric. Moreover, the symmetry is preserved if this similarity transformation is applied to arbitrary (rather than only Hermitian) Toeplitz matrices and even if it is applied to a much broader class of persymmetric matrices. Let the same similarity transformation be applied to the class of normal Toeplitz matrices. By examining the range of this transformation, commutative algebras are selected that consist of (complex) symmetric $(T + H)$-matrices; in addition, all the matrices in these algebras are normal. An algorithm is proposed for multiplying matrices belonging to these algebras. Its complexity is equivalent to that of multiplying two circulants of order $n$, which is several times less than the complexity of multiplying two general $(T + H)$-matrices.