  We define a linear functional, the DOS functional, on spaces of holomorphic
functions on the unit disk which is associated with random ergodic contraction
operators on a Hilbert space, in analogy with the density of state functional
for random self-adjoint operators. The DOS functional is shown to enjoy natural
integral representations on the unit circle and on the unit disk. For random
contractions with suitable finite volume approximations, the DOS functional is
proven to be the almost sure infinite volume limit of the trace per unit volume
of functions of the finite volume restrictions. Finally, in case the normalised
counting measure of the spectrum of the finite volume restrictions converges in
the infinite volume limit, the DOS functional is shown admit an integral
representation on the disk in terms of the limiting measure, despite the
discrepancy between the spectra of non normal operators and their finite volume
restrictions. Moreover, the integral representation of the DOS functional on
the unit circle is related to the Borel transform of the limiting measure.
