Consider the term "functional operator". My understanding was that:
(a) An operator in this context refers to a mapping from one vector space to another vector space.
(b) A functional is a mapping from a vector space to its underlying scalar field.
(c) Scalars are fundamentally different objects than vectors, and cannot, for instance, be thought of as little 1x1 vectors.
So if a functional is a mapping from a vector space into a scalar field, how can it be an operator? And I know that "operator" can sometimes have a much more general meaning ("something that does something") but I'm pretty sure in the context of a phrase like "functional operator" we're clearly in the more specific conversation about vector spaces.