Quasi-cyclic displacement and inversion decomposition of a quasi-Toeplitz matrix

We study a class of column upper-minus-lower (CUML) Toeplitz matrices, which are "close" to the Toeplitz matrices in the sense that their ( $1,-1$)-cyclic displacements coincide with $\phi$-cyclic displacement of some Toeplitz matrices. Among others, we derive the inverse formula for CUML Toeplitz matrices in the form of sums of products of factor circulants by constructing the corresponding displacement of the matrices. In addition, by the relationship between CUML Toeplitz matrices and CUML Hankel matrices, the inverse formula for CUML Hankel matrices is also obtained.