Volterra积分方程及其算子线性性的代数研究
An algebraic study of Volterra integral equations and their operator linearity
AI总结：
本文从代数角度研究Volterra积分算子与方程，构造匹配扭曲Rota-Baxter代数及自由对象，并证明可分离核积分方程具有算子线性性。
摘要：
对特殊积分算子的代数研究引出了Rota-Baxter算子与shuffle积的概念，它们已获得广泛应用。本文对一般积分算子和积分方程进行代数研究，并表明Volterra积分算子及相应方程背后存在丰富的代数结构。首先，证明了Volterra积分算子产生一个匹配扭曲Rota-Baxter代数，该代数满足扭曲的分部积分算子恒等式。为了提供表达一般积分方程的通用空间，随后利用带括号的字以及在顶点和边上带装饰的根树构造了自由算子代数。通过shuffle积的扭曲且带装饰的推广，进一步得到了匹配扭曲Rota-Baxter代数范畴中自由对象的显式构造，从而为可分离Volterra方程提供了通用空间。作为这些代数构造的应用，证明了任何具有可分离Volterra核的积分方程都是算子线性的，即该方程可以化简为迭代积分的线性组合。
英文摘要：
The algebraic study of special integral operators led to the notions of Rota-Baxter operators and shuffle products which have found broad applications. This paper carries out an algebraic study of general integral operators and equations, and shows that there are rich algebraic structures underlying Volterra integral operators and the corresponding equations. First Volterra integral operators are shown to produce a matching twisted Rota-Baxter algebra satisfying twisted integration-by-parts operator identities. In order to provide a universal space to express general integral equations, free operated algebras are then constructed in terms of bracketed words and rooted trees with decorations on the vertices and edges. Further explicit constructions of the free objects in the category of matching twisted Rota-Baxter algebras are obtained by a twisted and decorated generalization of the shuffle product, providing a universal space for separable Volterra equations. As an application of these algebraic constructions, it is shown that any integral equation with separable Volterra kernels is operator linear in the sense that the equation can be simplified to a linear combination of iterated integrals.
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