Permutability of Toeplitz and Hankel matrices

Conditions under which two Toeplitz matrices commute are known from at least 1998. For a pair of Hankel matrices, the corresponding commutativity conditions were recently obtained by Gel'fgat. To our knowledge, the problem of characterizing matrix pairs (T,H)(T,H) such that T is a Toeplitz matrix, H   is a Hankel matrix, and TH=HTTH=HT was so far not examined. We give a survey of easily found solutions to this problem and then conduct a detailed analysis of the rather intricate case where neither T nor H are centrosymmetric matrices. It turns out that, except for a very special situation, this case yields no new solutions to our permutability problem.