The spectrum of seminormal operators

With every pair of bounded self-adjoint operators {U,V} on Hilbert space such that VU - UV = (1/pii)C, where C is trace class, there is associated a certain function of two complex variables called the determining function of the pair. It was previously shown how the determining function can be obtained as the solution of a certain Riemann-Hilbert problem canonically associated with the pair, and how the complete spectral multiplicity theory for both U and V can be obtained from the determining function. We now show, under the condition that C is semidefinite, that the determining function method leads to a simple characterization of the spectrum of the seminormal operator T = U + iV.