Legal Question in Business Law in California

None. As long as there is no attempt or intent to defraud by the use of multiple names, you are free to do so, and many people do. For example it is extremely common for women who have established a professional name for themselves but have traditional values to keep their maiden name in business but take their husband's name for all other purposes. There is no legal proceeding required.

So long as you aren't cheating or defrauding anyone, you can legally call yourself anything you want. If you acquire real estate under one name and are using another at the time you sell, your deed must make reference to the name under which you went on title. Getting or renewing a passport will require satisfactory proof of your identity, and I'd suggest you consult with the passport agency on that one.