Inverses of infinite sign regular matrices

Let A be an infinite sign regular (sr) matrix which can be viewed as a bounded linear operator from l_∞ to itself. It is proved here that if the range of A contains the sequence ( … ,1, - 1,1, - 1, … ), then A is onto. If A- 1 exists, then DA- 1D is also sr, where D is the diagonal matrix with diagonal entries alternately $1$ and $- 1$. In case A is totally positive (tp), then DA- 1D is also tp under additional assumptions on A.