Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite sections (Q5943636)

It is considered the \(C^{\ast}\)-algebra of band-dominated operators with operator-valued coefficients. Fredholmness conditions for a class of band-dominated operators (discrete convolutions) are given in terms of limit operators. Local invertibility at infinity of such band-dominated operators is shown. Applicability of the finite section method is discussed. The \(C^{\ast}\)-subalgebra of \({\mathcal L}(L^2({\mathbb R}^n))\) containing some important classes of operators is introduced. It is shown that the operators in this \(C^{\ast}\)-subalgebra are unitarily equivalent to discrete band-dominated operators with operator-valued coefficients. The results are applied to restrictions of pseudodifferential operators onto cones in \({\mathbb R}^n\).