Pairs of operators with equal defect from unitarity and their relative index

We show that the $K_1$ group of a $C^*$-algebra $A$ can be defined as the homotopy classes of pairs of matrices over $A$ that have equal defect from being unitary. We also consider pairs of pseudodifferential operators, not necessarily elliptic, with symbols forming a balanced pair. A relative index is defined for such pairs of operators, and it is proved to be equal to the topological index of the pair of symbols.
Bibliography: 4 titles.