From scalar rigidity to abstract prime set and abstract integer system

We propose a unified framework for Prime Rigidity Theory (PR) by introducing the notions of an "abstract integer system" and an "abstract prime set", consisting, respectively, of two unbounded sequences $\mathcal{P}$ and $\mathcal{N}$ of elements of $\mathcal{X}$, where $\mathcal{X}$ denotes the algebra of bounded linear endomorphisms of a complex Banach space $X$, such that every element of $\mathcal{N}$ can be expressed as a finite product of powers of elements of $\mathcal{P}$. We define the zeta function $\zeta_{X,\mathcal{N}}$ on $\mathcal{X}\setminus{I_X}$, where $I_X$ is the identity endomorphism. The scalar restriction $\zeta_{\mathbb{C},\mathcal{N}}$ yields generalized Beurling zeta functions; in particular, $\zeta_{\mathbb{C},\mathbb{N}}$ coincides with the classical Riemann zeta function. We introduce the notions of a "rigid abstract integer system" and "rigid abstract prime set" in order to preserve the rigid asymmetric structure obtained in the scalar case.