The main evidence that could be used to make this argument is the evidence of muder rates in states that do and do not have the death penalty. If the death penalty deters people from murdering one another, we would expect that states that have the death penalty would have lower murder rates than states that do not. However, this is simply not the case. Murder rates are actually somewhat lower in states that do not have the death penalty.
Of course, this evidence is by no means conclusive. There are many ways to argue that this fact does not prove the deterrence perspective is wrong. However, this is the most compelling piece of evidence to show that the death penalty does not deter people from committing murder.