  Two different versions of an optical theorem for a scattering body embedded
inside a lossy background medium are derived in this paper. The corresponding
fundamental upper bounds on absorption are then obtained in closed form by
elementary optimization techniques. The first version is formulated in terms of
polarization currents (or equivalent currents) inside the scatterer and
generalizes previous results given for a lossless medium. The corresponding
bound is referred to here as a variational bound and is valid for an arbitrary
geometry with a given material property. The second version is formulated in
terms of the T-matrix parameters of an arbitrary linear scatterer circumscribed
by a spherical volume and gives a new fundamental upper bound on the total
absorption of an inclusion with an arbitrary material property (including
general bianisotropic materials). The two bounds are fundamentally different as
they are based on different assumptions regarding the structure and the
material property. Numerical examples including homogeneous and layered
(core-shell) spheres are given to demonstrate that the two bounds provide
complimentary information in a given scattering problem.
