The inverse of a totally positive bi-infinite band matrix

It is shown that a bounded bi-infinite banded totally positive matrix $A$ is boundedly invertible iff there is one and only one bounded sequence mapped by $A$ to the sequence $({( - )^i})$. The argument shows that such a matrix has a main diagonal, i.e., the inverse of $A$ is the bounded pointwise limit of inverses of finite sections of $A$ principal with respect to a particular diagonal; hence $({( - )^{i + j}}{A^{ - 1}}(i,j))$ or its inverse is again totally positive.