We use Operator Ideals Theory and Gershgorin theory to obtain explicit information in terms of the symbol concerning the spectrum of pseudo-differential operators, on a smooth manifold Ω with boundary $\partial \mathrm{\Omega }$, in the context of the non-harmonic analysis of boundary value problems introduced in [29] in terms of a model operator $\mathfrak{L}$. For symbols in the Hörmander class $S_{1,0}^0(\overline{\mathrm{\Omega }}\times \mathcal{I})$, we provide a non-harmonic version of Gohberg's lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is a compact operator in $L^2(\mathrm{\Omega })$, or a Riesz operator in $L^p(\mathrm{\Omega })$ in the case of Riemannian manifolds with smooth boundary. We extend to the context of the non-harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations.

中文翻译：

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非调和 Gohberg 引理、Gershgorin 理论和带边界流形上的热方程

我们使用算子理想理论和 Gershgorin 理论来获得关于伪微分算子谱的符号的明确信息，在一个光滑的流形 Ω 有边界 $\partial \mathrm{\Omega }$，在[29]中引入的基于模型算子的边值问题的非调和分析的背景下 $\mathfrak{升}$. 对于 Hörmander 类中的符号${秒}_{1,0}^0(\overline{\mathrm{\Omega }}\times \mathcal{一世})$，我们提供了一个非调和版本的 Gohberg 引理，以及一个充要条件来保证对应的伪微分算子是紧算子 ${升}^2(\mathrm{\Omega })$, 或 Riesz 算子 ${升}^{磷}(\mathrm{\Omega })$在具有光滑边界的黎曼流形的情况下。我们将关于椭圆算子精确域的众所周知的定理扩展到边值问题的非调和分析的上下文中，并讨论了所得结果在演化方程中的一些应用。