Looking at the photos, I would suggest that on the silver one, the handle has been sawn off to reveal a stump which has then been machined to a size for the silver handle to slide onto, or unlikely, it could have been threaded! I would go along with the heat suggestions whilst turning the handle anti clock wise in case it is threaded. Hold the black part in a vice protected with some pieces of wood, and apply some force. It may need plenty. As we say in Yorkshire, 'Give it some lad'. Remove the handle/knob, and place an extention tube over the handle to give extra leverage.
Jonty