Rigidity and Spectral Asymmetry for Operators in Linear Dynamics

We propose a generalisation programme for the rigidity theory developed in [1] to the setting of complex Banach spaces. he complex parameter $s = \sigma + i\tau$ is replaced by a bounded linear operator $W = A + B$, where $A$ is self‑adjoint and $B$ is anti‑Hermitian ($B^* = -B$). The symmetry $s \mapsto 1-\overline{s}$ becomes the operator involution $W \mapsto I - W^*$, and the scalar differential equation is lifted to an operator‑valued equation. We project the exact structure of [1], indicating the natural operator analogues of every object. The present text is a blueprint: Any computational error can be recovered by referring to the scalar proofs in [1].