Operators with a non-trivial closed invariant affine subspace

We are concerned with the question of the existence of an invariant proper affine subspace for an operator A on a complex Banach space. It turns out that the presence of the number 1 in the spectrum of A or in the spectrum of its adjoint operator \(A^*\) $A^{\ast }$ is crucial. For instance, an algebraic operator has an invariant proper affine subspace if and only if 1 is its eigenvalue. For an arbitrary operator A, we show that it has an invariant proper hyperplane if and only if 1 is an eigenvalue of \(A^*\) $A^{\ast }$ . If A is a power bounded operator, then every invariant proper affine subspace is contained in an invariant proper hyperplane, moreover, A has a non-trivial invariant cone.