Non--Vanishing functions and Toeplitz Operators on Tube--Type Domains

We prove an index theorem for Toeplitz operators on irreducible tube–type domains and we extend our results to Toeplitz operators with matrix symbols. In order to prove our index theorem, we proved a result asserting that a non–vanishing function on the Shilov boundary of a tube–type bounded symmetric domain, not necessarily irreducible, is equal to a unimodular function defined as the product of powers of generic norms times an exponential function.