Peripheral eigenspace dimensions for positive operators and semigroups

Let $T$ be a positive contraction on a monotonically complete complex Banach lattice with the Fatou property. We prove $$\dim\ker(\lambda I-T)\leq\dim\ker(\lambda^k I-T) \qquad (|\lambda|=1,\ k\in\mathbb Z),$$ where all infinite dimensions are identified with infinity. This answers an open problem posed by Jochen Glück in 2021 and extends the previously established root-of-unity case. The bounded bilateral trajectories of $T$ form an intrinsic Banach lattice by the fixed-space theorem. On this lattice, the shift is a lattice automorphism, and evaluation at zero identifies its unimodular eigenspaces with those of $T$. We also prove the corresponding multiplicity inequality for imaginary generator eigenspaces of positive contractive $C_0$-semigroups, and for positive power-bounded operators that are order continuous.

GitHub repository: https://github.com/eulogica/peripheral-eigenspace-cyclicity