The OPERATED BY was put there for FAA inspectors to do security checks, not to allow the Certificate Holder to use names of other entities, unless that other DBA was also listed as a DBA in the A001 OPSPEC.
For instance, "Delta Connection" operated by Atlantic Southeast Airlines. Atlantic Southeast Airlines was the certificate holder's name listed in A001, and "Delta Connection" was the DBA also listed in A001
This allowed several certificate holders to operate under code share agreements using the same marketing name, but all certificate holders were prohibited from holding out to the public as Delta Connection unless they also had such authorization as a DBA under A001.
If you are holding out the public using a business name, that name needs to be in OPSPEC A001 authroizing you to do so!
That is a black and white requirement of OPSPEC A008