Stephen Hawking's theory of black hole evaporation, known as Hawking radiation, is a theoretical prediction based on the principles of quantum mechanics and general relativity. According to this theory, black holes are not entirely black but emit a form of radiation over extremely long periods of time.
Hawking radiation arises from the quantum effects near the event horizon of a black hole. Virtual particle-antiparticle pairs continuously pop into existence in empty space, and normally, they annihilate each other shortly afterward. However, near the event horizon, it is possible for one of the particles to fall into the black hole while the other escapes into space. This escaping particle is perceived as radiation coming from the black hole, and it carries away energy, causing the black hole to lose mass over time.
The evaporation process is very slow for stellar-mass black holes and would take an extremely long time to complete. However, for supermassive black holes, which reside at the centers of galaxies, the evaporation timescale is much longer than the current age of the universe. This means that the Hawking radiation emitted by these black holes is extremely weak and negligible compared to other astrophysical processes.
Elliptical galaxies indeed have supermassive black holes at their centers, and the presence of these black holes is not in conflict with the theory of black hole evaporation. The formation and growth of supermassive black holes are complex processes that occur over cosmic timescales. While Hawking radiation will eventually cause black holes to lose mass, the timescales involved are so vast that the negligible amount of radiation emitted by these black holes has no significant impact on their growth or the galaxies they reside in.
It's important to note that our current understanding of black hole evaporation is based on theoretical models, and the direct observational confirmation of Hawking radiation from black holes is still a topic of ongoing scientific research.