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 $A^* : X \to X$ analogous to Hilbert space adjoints is avoided.

On the other hand, if $A:X \to Y$, the Banach space adjoint defined as $A^* : Y^* \to X^*$ 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.