Abstract

In this paper, we study partial integral operators on Banach–Kantorovich spaces over a ring of measurable functions. We obtain a decomposition of the cyclic modular spectrum of a bounded modular linear operator on a Banach–Kantorovich space in the form of a measurable bundle of spectra of bounded operators on Banach spaces. The classical Banach spaces with mixed norm are endowed with the structure of Banach–Kantorovich modules. We use such representations to show that every partial integral operator on a space with a mixed norm can be represented as a measurable bundle of integral operators. In particular, we show the cyclic compactness of such operators and, as an application, prove the Fredholm \(\nabla\)-alternative. We also give an example of a partial integral operator with a nonempty cyclically modular discrete spectrum, while its modular discrete spectrum is an empty set.

中文翻译：

---

Banach-Kantorovich 空间上的偏积分算子

摘要

在本文中，我们研究了可测函数环上的 Banach-Kantorovich 空间上的偏积分算子。我们以 Banach 空间上有界算子谱的可测丛的形式获得了 Banach-Kantorovich 空间上有界模线性算子的循环模谱的分解。混合范数的经典Banach空间被赋予了Banach-Kantorovich模的结构。我们使用这样的表示来表明具有混合范数的空间上的每个偏积分算子都可以表示为可测量的积分算子束。特别是，我们展示了此类算子的循环紧性，并作为一个应用，证明了 Fredholm \(\nabla\)-选择。我们还给出了一个具有非空循环模离散谱的偏积分算子的例子，而它的模离散谱是一个空集。