AI 中文总结
研究LCA群酉表示平均值的谱，通过证明\(\sigma(V) \subset \overline{\hat\mu(\Gamma)}\)及等式条件，利用谱测度证明谱映射定理，得出\(\mathbb{Z}\)和\(\mathbb{R}\)酉表示下\(\sigma(V)\)的具体结果。
AI 中文摘要
设\(G\)为具有对偶群\(\Gamma\)的局部紧阿贝尔（LCA）群，\(\mu\)为\(G\)上的概率测度。给定复希尔伯特空间\(H\)中的酉表示\(\{U(t): t \in G\}\)，研究\(\mu\) - 平均值\(V:=\int_G U(t)d\mu(t)\)（在\(H\)的强拓扑中定义）的谱。证明了\(\sigma(V) \subset \overline{\hat\mu(\Gamma)}\)并给出等式成立的充分条件。利用一般斯通定理给出的谱测度\(E(\cdot)\)，证明了关于算子\(U(\nu):=\int_GU(t)d\nu(t)\)的（弱）谱映射定理，其中\(\nu\)是\(G\)上的任意有界复测度。对于由酉算子\(U\)的幂定义的\(\mathbb{Z}\)的酉表示，证明了\(\sigma(V)={\widehat{\mu}}(\sigma(U))\)；对于给定为\(U(t)={\rm e}^{itB}\)（\(t\in\mathbb{R}\)）的\(\mathbb{R}\)的酉表示，表明\(\sigma(V)=\overline{\widehat{\mu}(\sigma(B))}\)。
英文摘要
Let $G$ be a locally compact Abelian (LCA) group with dual group $Γ$, and let $μ$ be a probability measure on (the Borel sets of) $G$. Given a unitary representation $\{U(t): t \in G\}$ in a complex Hilbert space $H$, we study the spectrum of the $μ$-average $V:=\int_G U(t)dμ(t)$ (defined in the strong topology of $H$). We prove that $σ(V) \subset \overline{\hatμ(Γ)}$ and give a sufficient condition for equality. Using the spectral measure $E(\cdot)$ given by the general Stone theorem, we prove a (weak) spectral mapping theorem for the operators $U(ν):=\int_GU(t)dν(t)$, where $ν$ is any bounded complex measure on $G$. For a unitary representation of $\mathbb Z$, defined by the powers of a unitary operator $U$, we prove that $σ(V)={\widehatμ}(σ(U))$. For a unitary representation of $\mathbb R$, given as $U(t)={\rm e}^{itB}$ ($t\in\mathbb R$), we show that $σ(V)=\overline{{\widehatμ}(σ(B))}$.
