The paper is devoted to the notion of a spectral section introduced by Melrose and Piazza. In the first part of the paper we generalize results of Melrose and Piazza to arbitrary base spaces, not necessarily compact. The second part contains a number of special cases, including cobordism theorems for families of Dirac type operators parametrized by a non-compact base space. In the third part of the paper we investigate whether Riesz continuity is necessary for existence of a spectral section or a generalized spectral section. In particular, we show that if a graph continuous family of regular self-adjoint operators with compact resolvents has a spectral section, then the family is Riesz continuous.

这篇论文主要讨论梅尔罗斯和皮亚扎提出的光谱截面的概念。在本文的第一部分中，我们将 Melrose 和 Piazza 的结果推广到任意基础空间，不一定是紧凑的。第二部分包含许多特殊情况，包括由非紧基空间参数化的狄拉克型算子族的配边定理。在本文的第三部分中，我们研究了里斯连续性对于谱截面或广义谱截面的存在是否是必要的。特别是，我们证明，如果具有紧解的正则自伴算子的图连续族具有谱部分，则该族是 Riesz 连续的。