Invertibility of a tridiagonal operator with an application to a non-uniform sampling problem

Let T be a tridiagonal operator on ℓ2(ℕ) which has strict row and column dominant property except for some finite number of rows and columns. This matrix is shown to be invertible under certain conditions. This result is also extended to double infinite tridiagonal matrices. Further, a general theorem is proved for solving an operator equation Tx = y using its finite-dimensional truncations, where T is a double infinite tridiagonal operator. Finally, it is also shown that these results can be applied in order to obtain a stable set of sampling for a shift-invariant space.