There are two schools of thought on the subject.
for it and against it.
I believe algae is better controlled by blocking sun from the nutrient solution. I also don't believe you should rely on H2O2 as your plants source for oxygen.
That being said, I do believe it can help (to an extent) those two issues.

Yeah, you shouldn't have algae if light is blocked. Otherwise, it's inevitable.

Peroxide is great for a lot of things, like killing powdery mildew and fungus, as well as disinfecting equipment. The problem with putting it in your reservoir is that peroxide is to bacteria what sevin dust is to insects - it kills everything, good and bad. If you have a well-built system with a highly oxygenated area for the roots to live, then your bacteria balance would be tilted toward the positive, and peroxide would hurt more than help since it would kill your beneficial bacteria. However, if you have brown, rotting roots, then you have an imbalance of bad bacteria, typically pythium. In that case, the peroxide would help by killing more bad than good. But that still is not really fixing whatever made your roots rot in the first place. It's just first-aid for a sick plant.

ConnR
Basically I agree with both grizzman and Cole here. H2O2 does help to increase dissolved oxygen levels in the water. But only temporarily. That's the biggest problem there. Using mechanical means (a pump, air/water) to regularly increase the dissolved oxygen level will help. But the type of system your growing in makes a big difference in how you set it up, and/or witch will work best for your situation.

H2O2 dissipates so quickly, you need to re-add it every other day. In cases of fungal/root disease it can defiantly help kill the pathogens ASAP, and keep it in check when used regularly. But is not a cure even then. Though it can be cheep insurance to kill pathogens when used regularly. But as cole mentions, it will kill both beneficial microbes as well as harmful ones (pathogens).

In the case of algae, like grizz mentions all you need to do is block light. If you do so you will not only kill 100% of the algae, but prevent it from reacquiring at all. Without light, algae can not exist. If you are still getting a growth in your water even though you blocked 100% of the light, it's not algae. It would be a fungal or bacterial growth then, and usually has a bad smell to it.

P.S.
Rotting roots are a food source for thousands of pathogens. Once pathogens establish themselves, their almost imposable to get rid of. Your best chance then is to regularly introduce beneficial microbes.