General almost automorphy

We will propose a very general setting for the following proposition. If a linear operator Lf is almost automorphic and all bounded solutions of the homogeneous equation Lg = 0 are almost automorphic then, for almost automorphic phi, all bounded solutions of the inhomogeneous equation Lf = phi are also almost automorphic.The almost automorphy previously introduced by us will be generalized and diversified into a property which we will name "A-invariance"; and we will deal with both strong solutions and weak solutions. We will also compare our A-invariance with minimality, and propose a rather thinned-out almost periodicity that will be associated with our A-invariance.