Why spectral norms are used for computing the complexity of adiabatic Hamiltonian?
The spectral norm $||H||$ determines the maximum energy involved in driving the evolution of the quantum system and thus the quantum computation. Any quantum evolution could be sped-up by a factor of $\lambda$ by simply driving the system with Hamiltonian $\lambda H$ as a simple consequence of Schrödinger's equation. The energy involved is simply another physical resource invested into solving the problem, that needs to be considered in complexity considerations, besides time. For example, if your Hamiltonian has a tiny spectral gap, you could of course enlarge it by simply rescaling the strengths of all interactions in your system until the gap is large enough in absolute terms. Of course, after having solved the engineering challenges involved, this will linearly accelerate your computationion, similar to overclocking your classical PC. But the speed gained was only achieved by investing another resource that needs to be kept track of.