  Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator
self-similar and (iii) stationary increment processes. They are the natural
multivariate generalizations of the well-studied fractional Brownian motions.
Because of the possible lack of time-reversibility, the defining properties
(i)--(iii) do not, in general, characterize the covariance structure of OFBMs.
To circumvent this problem, the class of OFBMs is characterized here by means
of their integral representations in the spectral and time domains. For the
spectral domain representations, this involves showing how the operator
self-similarity shapes the spectral density in the general representation of
stationary increment processes. The time domain representations are derived by
using primary matrix functions and taking the Fourier transforms of the
deterministic spectral domain kernels. Necessary and sufficient conditions for
OFBMs to be time-reversible are established in terms of their spectral and time
domain representations. It is also shown that the spectral density of the
stationary increments of an OFBM has a rigid structure, here called the
dichotomy principle. The notion of operator Brownian motions is also explored.
