Laconicity and redundancy of Toeplitz matrices

The convergence field of a Toeplitz matrix is a monotonic function of the set of rows that compose the matix, in the sense that the deletion of some of the rows of the matrix (followed by appropriate renumbering of the rows that remain) can never decrease the convergence field. In the case of certain matrices, the deletion of infinitely many rows always increases the convergence field; but there exist matrices that do not have this property. This dichotomy was considered with special reference to the space of bounded sequences and certain classical families of matrices.