This is pretty easy for you to test. Camera shops sell lightmeters, or if you attend a major university or know someone who does you can probably track down a truly sensitive lightmeter. After you do that just get ahold of a strong flashlight-
Measure the light from the flashlight directly, note it. In the same room and atmospheric conditions (I suggest you make sure your humidity levels are the same and there are no new dust particles in the air for both phases of the test) measure the strenght of the light reflected off of a mirror. If the instrument is sensitive enough you will note a diminished reflection. There is no such thing as a perfect mirror, in fact all a conventional mirror does is reflect along the visible spectrum plus and minus a bit.
Visible light changes as it loses energy. If you were to speed away from tyhe sun at near lightspeed you would soon outrun the visible spectrum you are used to and begin to see by other wavelenghts. While energy cannot be destroyed, it can and usually is converted to other formns of energy. Radiant energy is what you would get inside your sphere, as the light slowed it would become infra-red, violet, etc on down the line. Depending on how much light you had to begin with you will end up with some increase in heat, but you won’t get some ever increasing heat globe unless you continue to sent in light to be transformed to heat energy. A flashbulb would make only so much heat after it changed, no amount of reflection is going to multiply that.
Look up “red-shift” and “doppler effect visual spectrum”. I read somewhere that if we were to travel at those near C speeds then we may become blind or we may find ourselves seeing by radiowaves or other bands higher up the spectrum.