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 ${l_\infty }$ to itself. It is proved here that if the range of $A$ contains the sequence $( \ldots ,1, - 1,1, - 1, \ldots )$, then $A$ is onto. If ${A^{ - 1}}$ exists, then $D{A^{ - 1}}D$ is also sr, where $D$ is the diagonal matrix with diagonal entries alternately $1$ and $- 1$. In case $A$ is totally positive (tp), then $D{A^{ - 1}}D$ is also tp under additional assumptions on $A$.