  We consider the linearized instability of 2D irrotational solitary water
waves. The maxima of energy and the travel speed of solitary waves are not
obtained at the highest wave, which has a 120 degree angle at the crest. Under
the assumption of non-existence of secondary bifurcation which is confirmed
numerically, we prove linear instability of solitary waves which are higher
than the wave of maximal energy and lower than the wave of maximal travel
speed. It is also shown that there exist unstable solitary waves approaching
the highest wave. The unstable waves are of large amplitude and therefore this
type of instability can not be captured by the approximate models derived under
small amplitude assumptions. For the proof, we introduce a family of nonlocal
dispersion operators to relate the linear instability problem with the elliptic
nature of solitary waves. A continuity argument with a moving kernel formula is
used to study these dispersion operators to yield the instability criterion.
