Operators with a non-trivial closed invariant affine subspaceBračič, Janko (Avtor) invariant subspacesinvariant affine subspacespower bounded operatorsWe 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 $A^*$ 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 $A^*$. 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.20242025-01-10 11:41:20Članek v reviji166410UDK: 517.9ISSN pri članku: 0001-9054DOI: 10.1007/s00010-024-01090-0COBISS_ID: 221338371sl