Surjectivity and invertibility properties of totally positive matrices

A new “finite section” type theorem is used to show that the members of an interesting class of bounded totally positive matrices map l_∞ onto l_∞ if and only if their range contains a vector which alternates in sign and has coordinates bounded away from zero. The class of matrices studied contains all banded totally positive matrices, and thus all infinite spline collocation matrices. Connections to related work and extension to matrices which are not sign regular are indicated.