部分等距算子的Schatten类扰动的谱特征
Spectral Characterizations of Schatten-class perturbations of Partial isometries
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中文总结 AI 辅助
研究刻画部分等距算子标量倍数的紧或Schatten类扰动的有界算子，通过\(T^*T\)本质谱等进行，给出相关充要条件及等价准则，还应用于等距算子扰动刻画、Moore - Penrose逆行为描述及闭值域算子分解。
中文摘要 AI 辅助
我们刻画了有界算子，这些算子是具有有限维核的部分等距算子的标量倍数的紧（分别为Schatten类）扰动。我们通过\(T^*T\)的本质谱、绝对范数达到算子和Moore - Penrose逆来进行刻画。特别地，我们表明算子\(T\)是具有有限维核的部分等距算子的Schatten类扰动，当且仅当\(\sigma_{\mathrm{ess}}(T^*T)\)是单元素集且\(T^*T\)的离散谱满足相应的\(\ell^p\)可和性条件。我们还得到了涉及\(\alpha I - T^*T\)和\(\alpha T^\dagger - T^*\)的紧性（或Schatten类成员资格）的等价准则。作为应用，我们建立了等距算子的紧和Schatten类扰动的刻画，描述了Moore - Penrose逆的相应行为，并推导了闭值域算子的分解结果。特别地，我们为Şerban和Turcu的一个定理提供了新的Moore - Penrose逆证明，并得到了分解算子的显式公式。
英文摘要
We characterize bounded operators that are compact (respectively, Schatten-class) perturbations of scalar multiples of partial isometries with finite-dimensional kernel. Our characterizations are formulated in terms of the essential spectrum of $T^*T$, absolutely norm attaining operators, and the Moore-Penrose inverse. In particular, we show that an operator $T$ is a Schatten-class perturbation of a partial isometry with finite-dimensional kernel if and only if $σ_{\mathrm{ess}}(T^*T)$ is a singleton and the discrete spectrum of $T^*T$ satisfies a corresponding $\ell^p$-summability condition. We further obtain equivalent criteria involving the compactness (or Schatten-class membership) of $αI-T^*T$ and $αT^\dagger-T^*$. As applications, we establish characterizations of compact and Schatten-class perturbations of isometries, describe the corresponding behavior of Moore--Penrose inverses, and derive factorization results for closed-range operators. In particular, we provide a new Moore--Penrose inverse proof of a theorem of Şerban and Turcu and obtain an explicit formula for the factorizing operator.
