An ELT Glossary : The Operator

Definition : The operator is the name given to the first auxiliary verb in a verb phrase, which is used for grammatical operations such as inversion, negation etc.

Examples : 
- in I'm going to the supermarket  the operator is be. The interrogative would be formed by inverting BE and the subject : Are you going to the supermarket?
- in I'll be going to the supermarket later the operator is will. It would eg carry the negative particle not in a negative sentence : I won't be going to the supermarket

When there is no auxiliary verb included in the affirmative verb phrase (ie with the simple present and past forms) then the auxiliary Do is used as a "dummy" operator.  It adds no meaning to the verb phrase, but is simply there to carry out the functions of the operator.