Classification of finite-dimensional algebras generated by the calkin image of a composition operator on $L^p$ with weight

Given a countable infinite set $X$ and a weight $\mu\colon X\to(0,\infty)$, we denote by $l_{\mu}^p(X)$ the Banach space of all functions $f\colon X\to\mathbb C$ such that $\sum_{x\in X}|f(x)|^p\mu(x)<\infty$. The composition operator $C_a$ on $l_{\mu}^p(X)$ induced by a self-map $a\colon X\to X$ is defined by $(C_af)(x)=f(a(x))$. We establish a criterion for $C_a$ to be essentially algebraic, i.e., for the existence of a polynomial $q(z)$ such that $q(C_a)$ is compact. The polynomial $q(z)$ of minimal degree with this property is referred to as the essentially characteristic polynomial of $C_a$. We provide a list of all polynomials that may be the essentially characteristic polynomial of some composition operator on $l_{\mu}^p(X)$, which results in a complete classification of the finite-dimensional algebras generated by the Calkin image of a single composition operator on $l_{\mu}^p(X)$.