On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Let $\mathbb{S}$ be a cone in ?ⁿ. A bounded linear operator T: L_p (?ⁿ) → L_p (?ⁿ) is said to be causal with respect to $\mathbb{S}$ if the implication x(s) = 0 (s ε W ? $\mathbb{S}$) ? (Tx) (s) = 0 (s ε W ? $\mathbb{S}$) is valid for any x ε L_p (?ⁿ) and any open subset W$\subseteq$ ?ⁿ. The set of all causal operators is a Banach algebra. We describe the spectrum of the operator

$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$

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 $\mathbb{E}$.