Meaning of A, A inverse Adjoint A and inverse of Adjoint A between two normed spaces

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I am trying to understand concept of Adjoint Operator from Functional Analysis by George Bachman. Following is an extract from page 354, of theorem 20.2. I am getting confused between $A$,$A^{-1}$,$A^{*}$,$(A^{-1})^{*}$,$D_{A}$, $D_{A^{*}}$. I have tried to draw this diagrammaticallyenter image description here, but without much success. Can anyone please help me redraw this diagram for better understanding?