I’ve read your previous question and the fine responses which provide the context for responding to this question of building trust. From the information available, it appears that the founder (and President) is now ready to delegate the day-to-day operations to you, his new Chief Operating Officer. However, it also appears that the founder has trained those who now report to you to “delegate up” to him. It is not part of the culture for anyone to do anything new or different without first consulting the founder. Your “presence” disrupts and interferes with long established, comfortable habits.
The organization operated successfully before you arrived. So in dealing with problems or operational changes, have your direct reports teach you how they have solved similar problems in the past and how they want to solve the current problem today. IF their idea will work, even if it is not as good as your idea, let them execute their approach. Introduce as few of your ideas as possible for at least six months. Once your direct reports feel that you respect and value their talents and experience, they will be open to learning from you.
The second aspect of building trust “is doing what you say you will do when you say you will do it, in all things large and small.” When you promise to deliver something such as a report, recommendation, or whatever, by a time and date certain, you deliver! You are “the new kid on the block,” so it is essential that they believe you are who you say you are and will do what you say you will do, when you say you will do it!
The third aspect of building trust requires vulnerability and transparency. I frame this as a “HOT” Relationship: Honesty and Openness builds Trust. If you are able to say “I don’t know, what do you think?” when you don’t know, you will more quickly build the trust required to be the leader they need and will follow. One of the most essential qualities of leadership today is the ability to really listen.