  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.
