An operator operates on a function to produce a new function.
There is a special situation where that new function is actually the product of a scalar and f(x): 
When this happens in Quantum Mechanics, we say that ‘a’ is an observable for whatever process the operator represents.
One cautionary note: to be an eigenfunction, a function has to be an eigenfunction of a particular operator. There is an interdependence between the two, and proving a function to be an eigenfunction for one operator doesn’t mean it will be an eigenfunction for all operators.
Rob Sterling 