If a company is operating under my web domain name, can I file a lawsuit?

If your domain name is your trademark, and the other company is in the same business/geographical area, you may have a basis to file a lawsuit for trademark infringement.

A company that is operating under a name that is identical to your web domain might be infringing your rights if you have acquired trademark rights in your web domain and if their use is confusingly similar to yours. It all comes down to the specific facts of each company and the history of operation of each company. This is an example of a time when you wish you had a federal trademark registration. If you had a federal trademark registration, your rights would be a lot more secure and when you notify the other party about the possible infringement, you have a lot better chance to get them to change their name.