Reduced spectral synthesis and compact operator synthesis

2019/01/17 by Victor S. Shulman, Ivan G. Todorov, Shulman, V. S. +3
Mathematics · #46A32 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47L05 #Secondary: 43A45 #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1901.05832

openalex publication_date 2019/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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∈ E\ of G× 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.

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