No, they are not. Don't believe me, check your state's statutes.
I don't care what the statutes say. When you let your wife and kids go to the mall together for instance (if you have any...), and they have to mingle with all the street urchins that congregate there, it's not the official "police force" that are there keeping an eye on them for you while you are not there. It's not a guy with a permenant badge, a cruiser, and a shotgun in the front seat who is responsible for them staying safe while they walk around inside and then carry all those Christmas presents across the dark parking deck on the way back to the safety of their vehicle. It's the mall security police who are there for your family while you are unable to be there protecting them.
It is mall security that will call the real police when something happens. Sorry, I have dealt with way to many "private security operators" in my life. There are some good and well intentioned ones, as I am sure this gentleman is, but there are many more who are just plain wackos or slugs. There is nothing wrong with having a security job, but that is exactly what it is, a security job. Let's not pretend it is something it is not.