we have a contractive analytic in the unit disk matrix-valued function ( ), which is unitary on the unit circle ( ), and which in nondegenerate throughout the unit disk. I want to conclude that is a constant unitary matrix. We can assume that is analytic in some neighbourhood of the unit disk (rather than analytic inside the unit disk with boundary values which are unitary)

the proof for the 1D case could go as follows -- since f doesn't vanish, we can take which is a harmonic function. on the unit circles is zero, so by maximum principle for harmonic functions it is zero everywhere, i.e. is constant everywhere, and then so is f. clearly this approach doesn't help me in the matrix case