D-Cyclic Operators: A Unified Framework for Cyclicity in Linear Dynamics

We introduce the notion of D-cyclicity for bounded linear operators on a separable infinite-dimensional complex Banach space, which unifies several classical cyclicity concepts through appropriate choices of D⊂C. We establish fundamental properties of D-cyclic operators, including stability under commuting surjective compositions and invariance under unimodular scalings. For invertible operators, we also show a correspondence between D-cyclicity of T and D−1-cyclicity of T−1, and we prove that the backward shift on ℓ2(N) is D-cyclic whenever D is unbounded. Moreover, when D is bounded and bounded away from zero, D-cyclic operators exhibit Li–Yorke chaos and satisfy spectral restrictions that exclude adjoint eigenvalues. Furthermore, we prove that if D contains a closed disk around the origin, then a D-cyclic operator admits no non-trivial closed invariant subspace of finite co-dimension.