AI 中文总结
研究具有不变子空间的压缩算子和行压缩算子特征函数分解的正则性，构造反例说明一般非正则，给出正则性刻画，确定几类正则算子，引入\(k -\)奇异分解概念并证明特定压缩算子特征函数不存在\(k -\)奇异分解。
AI 中文摘要
给定一个具有不变子空间的压缩算子和一个具有联合不变子空间的行压缩算子，Sz.-Nagy 和 Foiaş 以及 Haria、Maji 和 Sarkar 分别得到了它们特征函数的分解。本文研究了单变量和多变量情况下这种分解的正则性。构造了具有不变子空间的压缩算子\(T\)的例子，表明一般情况下这种分解不具有正则性。若\(T\)是完全非酉的，则证明这种分解要么正则要么奇异。还得到了这种分解正则性的一个刻画，借此确定了几类分解正则的压缩算子和行压缩算子，最显著的是纯压缩算子和纯行压缩算子。此外，引入了压缩解析函数的\(k -\)奇异分解概念，扩展了 Sz.-Nagy 和 Foiaş 引入的奇异（\(2 -\)奇异）分解概念。最后证明若一个压缩算子是纯的或完全非酉且\(\Delta_{\Theta_T}(t)\)几乎处处具有有限秩，则其特征函数不存在任何\(k -\)奇异分解。
英文摘要
Given a contraction with an invariant subspace and a row contraction with a joint invariant subspace, a factorization of their characteristic functions was obtained by Sz.-Nagy and Foiaş, and by Haria, Maji, and Sarkar, respectively. In this article, we investigate the regularity of this factorization in both the single-variable and multivariable cases. We construct examples of a contraction $T$ with an invariant subspace such that this factorization is not regular in general. If $T$ is completely non-unitary, we prove that this factorization is either regular or strange. Furthermore, we obtain a characterization of the regularity of this factorization. Using this characterization, we identify several classes of contractions and row contractions for which this factorization is regular. Among these classes, the most notable classes are pure contractions and pure row contractions. Additionally, for any integer $k>2$, we introduce the concept of $k$-strange factorizations for contractive analytic functions, which extend the concept of strange ($2$-strange) factorizations introduced by Sz.-Nagy and Foiaş. Finally, we prove that if a contraction is pure or is a completely non-unitary contraction for which $Δ_{Θ_T}(t)$ has finite rank almost everywhere, then its characteristic function does not admit any $k$-strange factorization.
Comments19 pages
