We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it with other exceptional sets in operator algebra theory, studied previously. We show that a closed subset $E$ of a second countable locally compact group $G$ satisfies reduced local spectral synthesis if and only if the subset $E^* = \{(s,t) : ts^{-1}\in E\}$ of $G\times G$ satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten $p$-classes.

中文翻译：

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减少频谱合成和紧凑算子合成

我们介绍并研究了简化谱合成的概念，它统一了局部紧群中的谱合成和唯一性的概念。我们展示了许多示例，并证明每个具有开放阿贝尔子群的非离散局部紧致群都有一个无法减少谱合成的子集。我们引入紧凑算子合成作为这个概念的算子代数对应物，并将其与之前研究过的算子代数理论中的其他例外集联系起来。我们表明，当且仅当子集 $E^* = \{(s,t) : ts^{-1}\ 时，第二个可数局部紧群 $G$ 的封闭子集 $E$ 满足减少的局部谱合成在 $G\times G$ 的 E\}$ 中满足紧凑算子综合。