  Let X_N= (X_1^(N), ..., X_p^(N)) be a family of N-by-N independent,
normalized random matrices from the Gaussian Unitary Ensemble. We state
sufficient conditions on matrices Y_N =(Y_1^(N), ..., Y_q^(N)), possibly random
but independent of X_N, for which the operator norm of P(X_N, Y_N, Y_N^*)
converges almost surely for all polynomials P. Limits are described by operator
norms of objects from free probability theory. Taking advantage of the choice
of the matrices Y_N and of the polynomials P we get for a large class of
matrices the "no eigenvalues outside a neighborhood of the limiting spectrum"
phenomena. We give examples of diagonal matrices Y_N for which the convergence
holds. Convergence of the operator norm is shown to hold for block matrices,
even with rectangular Gaussian blocks, a situation including non-white Wishart
matrices and some matrices encountered in MIMO systems.
