Trust is a two-way street. People do not need to prove trustworthy to be trusted. It is vice versa. Trust is given, then withdrawn if one proves not worthy of it. This does not mean you give all trust all at once to someone. It is given gradually, but the assumption is always that someone is worthy of trust until proven otherwise. Let me explain:
When you have a new employee working for you. He asks you to give him a chance to show you how he can do something in a better way. Depending on how much you “trust” his abilities, you might or might not allow him to do it.
Let us consider Scenario A, if the manager gives the employee a chance to do the work and s/he succeeds, trust rises. If the opposite, the manager might have less trust in the ability of the employee. So, trust is built over time and must be assumed first. In the same example, there is a Scenario B: if the manager feels that the employee needs to prove he is trustworthy first before giving trust, then the manager might lose a chance to get value out of the employee by denying the employee that trust.
The difference between the two scenarios, even if subtle on the surface, is a major difference. In scenario A you are open to benefit from the trust. In scenario B, you are closed to trust, and will only benefit from it from the few that proved to you already that they are trustworthy. Think about it: the people who you know for a fact are trustworthy are very few. However, the whole population out there is available for you to trust by default and benefit from that trust.
Here is another problem with withholding trust except for a few people. Let us face it, almost everybody are bound to disappoint you one day or the other. Does that mean they are not trustworthy? under this test, you will end up not trusting anyone. That is a gloomy state to be in.
Give trust, and allow people to disappoint you once in a while. You gain much more on the long run. Another thing I notice is that when you give people trust they rise to the occasion. When I expect team members to perform, they are more likely to succeed, because they want to rise to my expectations. But if I do not trust, then their feelings of responsibility and motivation and wanting to rise to your expectations are much less, as they know you do not trust them anyways. This is why the same team members might do excellent work under one manager, while do less than mediocre under another manager. In many cases the difference is in how much does a manager trust the abilities of the team member.
Give trust. Build it. Benefit from it.