  We explore the possibility of extending the well-known Berezin-Toeplitz
quantization to reproducing kernel spaces of vector-valued functions. In
physical terms, this can be interpreted as accommodating the internal degrees
of freedom of the quantized system. We analyze in particular the vector-valued
analogues of the classical Segal-Bargmann space on the domain of all complex
matrices and of all normal matrices, respectively, showing that for the former
a semi-classical limit, in the traditional sense, does not exist, while for the
latter only a certain subset of the quantized observables have a classical
limit: in other words, in the semiclassical limit the internal degrees of
freedom disappear, as they should. We expect that a similar situation prevails
in much more general setups.
