I know about them Its not what I mean. I'm talking about utilizing all the optics of an FX lens on a DX body.
I know. You are talking about an optic between the lens and the body which takes the full frame view of the lens and converges it a bit so it all fits onto a crop frame sensor.
A slightly convex lens will do this - but the image would then be out of focus so you would need to compensate by moving the lens closer to the sensor. Unfortunately the lens mount gets in the way.
I have done this with a large format camera which has the benefit of bellows and the ability to put the lens where you like. I used a 135mm lens with a +1 dioptre supplementary lens in front - it will work the same at the back (I tried that too).
What you are actually doing is making the lens a bit more wide angle than it was. In my example the 135mm lens is 7.4 dioptres (1/focal length in metres). Add the 1 dioptre of the extra lens = 8.4 dioptres. This makes the new focal length 119mm.
On a view camera with bellows, the lens is usually positioned at about the focal length away from the film. In the case of my 135mm lens it is around 135mm away from the film as expected. Add the supplementary lens and it needs to be a bit closer, probably at 119mm from the film just like a wider angle lens - which it has just become.
In my case, this converts the lens to give a wider view on the same format. This is the same thing as you are trying to achieve but in your case you are looking for the same image on a smaller format - but if you increased the format back to full frame, you would get a wider view.
So in summary, you could do it theoretically but you would need to remove the lens mount and move it all closer - and the mirror would probably hit it.
Steve.