On algebras of Toeplitz plus Hankel matrices (Q1894486)

Let \({\mathcal K}\) be the space of all \(n \times n\) matrices which are the sum of a Toeplitz and a Hankel matrix. The authors use properties of displacement operators to give a nice proof that \(A \in {\mathcal K}\) if and only if \(A\) satisfies the cross-sum condition \[ a_{i - 1,j} + a_{i + 1,j} = a_{i,j - 1} + a_{i,j + 1} \quad \text{for all } i,j = 2, \dots, n - 1 \] and to characterize a wide class of maximal algebras in \({\mathcal K}\).