Hi Bear,
In a purist sense, be it a fan-less jet engine or a turbo fan engine, the turbine or turbines drive the compressor or compressor/fan combination. The only time the compressor or compressor/fan combination "drives" is during the start operation. Here, the compressor or compressor/fan combination through an auxiliary energy source initiates rotation, driving the turbine/turbines while compressing the air delivered to the combustor. When the combustor delivered air pressure is sufficient, fuel flowing into the combustor is ignited. From that point onward, the turbine/turbines drive the compressor or compressor/fan combination.
Hope this clarifies this discussion.