Joint torsion of Toeplitz operators with H∞ symbols

We give a new(2) index theorem for the basic example of Toeplitz operators on the circle. The joint torsion, a non zero complex valued analytic index, of a pair of Fredholm Toeplitz operators T-phi and T-psi with H-infinity symbols is computed by residues in the disk, and is determined by a monodromy integral which specifies the isomorphism class of phi boolean OR psi a flat line bundle on the circle. When the symbols phi and psi are rational a product of joint torsions identifies the isomorphism class of the bundle in H-1(S-1, C*), and the identification extends by rational approximation to the case of smooth symbols defined on the circle.