Some TRVs do not have a positive OFF position. If the lowest setting is a *, that is a frost setting. What is your lowest setting?
A TRV will be fully open when the room temperature is about 2 degrees below the set temperature and slowly closes as the room temperature rises. It is never completely closed, just enough so that the flow through the rad is sufficient to supply the required amount of heat to maintain the room at the correct temperature.
The frost setting is slightly different. If the room temperature is above the frost setting, the TRV will be closed, But when the room temperature drops below the frost setting the TRV will open to allow water to flow through the rad and maintain the room at the frost temperature.