  Unbounded outflows in the form of highly collimated jets and broad winds
appear to be a ubiquitous feature of accreting black hole systems. The most
powerful jets are thought to derive a significant fraction, if not the
majority, of their power from the rotational energy of the black hole. Whatever
the precise mechanism that causes them, these jets must therefore exert a
braking torque on the black hole. We calculate the spin-up function for an
accreting black hole, accounting for this braking torque. We find that the
predicted black hole spin-up function depends only on the black hole spin and
dimensionless parameters describing the accretion flow. Using recent
relativistic magnetohydrodynamical numerical simulation results to calibrate
the efficiency of angular momentum transfer in the flow, we find that an ADAF
flow will spin a black hole up (or down) to an equilibrium value of about 96%
of the maximal spin value in the absence of jets. Combining our ADAF system
with a simple model for jet power, we demonstrate that an equilibrium is
reached at approximately 93% of the maximal spin value, as found in the
numerical simulation studies of the spin-up of accreting black holes, at which
point the spin-up of the hole by accreted material is balanced by the braking
torque arising from jet production. Our model also yields a relationship
between jet efficiency and black hole spin that is in surprisingly good
agreement with that seen in the simulation studies, indicating that our simple
model is a useful and convenient description of ADAF inflow - jet outflow about
a spinning black hole for incorporation in models of the formation and
evolution of galaxies, groups and clusters of galaxies.
