In this paper, a novel framework is proposed for optimizing the operation and performance of a large-scale, multi-hop millimeter wave (mmW) backhaul within a wireless small cell network (SCN) that encompasses multiple mobile network operators (MNOs). The proposed framework enables the small base stations (SBSs) to jointly decide on forming the multi-hop, mmW links over backhaul infrastructure that belongs to multiple, independent MNOs, while properly allocating resources across those links. In this regard, the problem is addressed using a novel framework based on matching theory that is composed to two, highly inter-related stages: a multi-hop network formation stage and a resource management stage. One unique feature of this framework is that it jointly accounts for both wireless channel characteristics and economic factors during both network formation and resource management. The multi-hop network formation stage is formulated as a one-to-many matching game which is solved using a novel algorithm, that builds on the so-called deferred acceptance algorithm and is shown to yield a stable and Pareto optimal multi-hop mmW backhaul network. Then, a one-to-many matching game is formulated to enable proper resource allocation across the formed multi-hop network. This game is then shown to exhibit peer effects and, as such, a novel algorithm is developed to find a stable and optimal resource management solution that can properly cope with these peer effects. Simulation results show that the proposed framework yields substantial gains, in terms of the average sum rate, reaching up to 27% and 54%, respectively, compared to a non-cooperative scheme in which inter-operator sharing is not allowed and a random allocation approach. The results also show that our framework provides insights on how to manage pricing and the cost of the cooperative mmW backhaul network for the MNOs.