In no case does the resident permit for property owners provide access to any type of employment. Employment, according to paragraph 6, article 20, of Law 4251/2014, does not include the exercise of economic activity in the capacity of a shareholder or a chief executive officer for an already existing company. So under the above roles you could proceed to a business activity.
How is it possible to start a company in Greece as a Golden Visa holder?
I recently obtained my Golden Visa for my family. I live in Athens and am interested in starting a business here. How can I do this, become the chief executive, and work under my Golden Visa status? Are there any limitations to the types of business a foreigner can operate?
As you already obtained the Greek Golden Visa, there are no limitations in relation to that procedure. You may start a business and select a company type that fits to that best in relation to ongoing taxation law applicable.
A Greek Golden Visa holder cannot work in Greece but he could set up a company as a shareholder or be a CEO.