On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces

Let $\mathbb{S}$ be a cone in $\mathbb{R}^n$. A bounded linear operator $T\colon L_p(\mathbb{R}^n)\to L_p(\mathbb{R}^n)$ is said to be causal with respect to $\mathbb{S}$ if the implication
$$ x(s)=0\;\;(s\in W-\mathbb{S})\implies(Tx)(s)=0\;\;(s\in W-\mathbb{S}) $$
is valid for any $x\in L_p(\mathbb{R}^n)$ and any open subset $W\subseteq\mathbb{R}^n$. The set of all causal operators is a Banach algebra. We describe the spectrum of the operator
$$ (Tx)(t)=\sum_{n=1}^\infty a_n x(t-t_n)+ \int_{\mathbb{S}}g(s)x(t-s)\,ds,\qquad t\in\mathbb{R}^n, $$
in this algebra. Here $x$ ranges in a Banach space $\mathbb{E}$, the $a_n$ are bounded linear operators in $\mathbb{E}$, and the function $g$ ranges in the set of bounded operators in $\mathbb{E}$.