in embry's 1973 and douglas' 1966 papers the hilbert space condition AA* < lambda^2 BB* about the adjoints is replaced by the following equivalent banach type of condition: ||A|| < lambda ||B||, so that the whole exotic question of banach space adjoints A*:X-->X analogous to hilbert space adjoints is avoided.
on the other hand, if A:X-->Y the banach space adjoint defined as A*:Y-->X using duality theory does play an important role in the results of douglas and embry. embry's is accessible on the free internet and is easy to read.