nLab
operator

There are different meanings of operator:

• A linear operator, or more generally an element of an operator algebra.

• In higher type theory?, an operator is a function from a set of functions to itself:

  $$O:Y^X\to Y^X,O\in (Y^X{)}^{Y^X}=Y^{X\times Y^X}.$$O\colon Y^X \to Y^X , \quad O \in (Y^X)^{Y^X} = Y^{X \times Y^X} .

• Probably more …

The connection between these two is that, if $H$ is a vector space of functions ($H\subseteq Y^X$), then a linear operator (in the first) is a special case of a (partially defined) operator (in the second sense).