Emissivity and absorptivity relation question This question is related to another discussion here about emissivity of bodies.  What interested me is to verify what extend the wavelength between emissivity and absorption is conserved at thermal equillibrium. That is , emissivity is produced only from electromagnetism with wavelengths range absorbed?
I hope the point of my question is clear. But just in case, another way to lay my question.  Is it possible for a narrow bandwith of radiation and x joules of energy center at a wavelength   produces the  emission with the same amount of energy but a bandwidth centered at a very different place  (up or down)?
 A: 
a narrow bandwith of radiation and x joules of energy center at a wavelength produces the emission with the same amount of energy but a bandwidth centered at a very different place 

If I understand correctly, you are asking about a greenhouse effect.   The
'narrow bandwidth' of incoming thermal radiation may be balanced by outgoing
radiation when a body is at radiative thermal steady state, but the
outgoing radiation is characterized by the temperature of the body,
NOT by the temperature of the source of incoming radiation, so will have different
spectral band character.   Both the temperature and the emissivity 
determine thermal radiant emission.
Earth, at a thermal steady state, shines thermally at IR and microwave
wavelengths, but is warmed mainly by solar (white/visible light) wavelengths.
The power in those two flows is in different spectral regions.
If Earth were really at equilibrium with Sol, it would be at the same
temperature; steady-state is a better name for the thermal resting
temperature between that of Sol and the black void.
