A commuting tuple of n operators $(S_1,\dots ,S_{n-1},P)$ defined on a Hilbert space $\mathcal{H}$, for which the closed symmetrized polydisc${\mathrm{\Gamma }}_n=\{(\sum _{i=1}^n{z}_i,\sum _{1\le i<j\le n}{z}_i{z}_j,\dots ,\prod _{i=1}^n{z}_i):|z_i|\le 1,i=1,\dots ,n\}$ is a spectral set is called a ${\mathrm{\Gamma }}_n$-contraction. Also a triple of commuting operators $(A,B,P)$ for which the closed tetrablock $\bar{\mathbb{E}}$ is a spectral set is called an $\mathbb{E}$-contraction, where$\mathbb{E}=\{(x_1,x_2,x_3)\in {\mathbb{C}}^3:1-z{x}_1-w{x}_2+zw{x}_3\ne 0\forall z,w\in \bar{\mathbb{D}}\}.$ There are several decomposition theorems for contraction operators in the literature due to Sz. Nagy, Foias, Levan, Kubrusly, Foguel and few others which reveal structural information of a contraction. In this article, we obtain analogues of six such major theorems for both ${\mathrm{\Gamma }}_n$-contractions and $\mathbb{E}$-contractions. In each of these decomposition theorems, the underlying Hilbert space admits a unique orthogonal decomposition which is provided by the last component P. The central role in determining the structure of a ${\mathrm{\Gamma }}_n$-contraction or an $\mathbb{E}$-contraction is played by positivity of some certain operator pencils and the existence of a unique operator tuple associated with a ${\mathrm{\Gamma }}_n$-contraction or an $\mathbb{E}$-contraction.

n个运算符的通勤元组$（{\mathrm{小号}}_{\mathrm{1个}}，\dots ，{\mathrm{小号}}_{ñ-\mathrm{1个}}，P）$ 在希尔伯特空间上定义 $\mathcal{H}$，其封闭的对称多碟${\mathrm{\Gamma }}_{ñ}=\{（\sum _{\mathrm{一世}=\mathrm{1个}}^{ñ}{ž}_{\mathrm{一世}}，\sum _{\mathrm{1个}\le \mathrm{一世}<Ĵ\le ñ}{ž}_{\mathrm{一世}}{ž}_{Ĵ}，\dots ，\prod _{\mathrm{一世}=\mathrm{1个}}^{ñ}{ž}_{\mathrm{一世}}）：|{ž}_{\mathrm{一世}}|\le \mathrm{1个}，\mathrm{一世}=\mathrm{1个}，\dots ，ñ\}$ 是一个光谱集称为 ${\mathrm{\Gamma }}_{ñ}$-收缩。也是三重通勤运营商$（\mathrm{一种}，乙，P）$ 封闭的四嵌段 $\stackrel{〜}{\mathbb{Ë}}$ 是一个光谱集称为 $\mathbb{Ë}$-收缩，在哪里$\mathbb{Ë}=\{（X_{\mathrm{1个}}，X_2，X_3）\in {\mathbb{C}}^3：\mathrm{1个}-žX_{\mathrm{1个}}-w{X}_2+žw{X}_3\ne 0\forall ž，w\in \stackrel{〜}{\mathbb{d}}\}。$由于Sz，在文献中有几个关于收缩算子的分解定理。Nagy，Foias，Levan，Kubrusly，Foguel和其他一些揭示收缩结构信息的人。在本文中，我们获得了两个这样的主要定理的类似物${\mathrm{\Gamma }}_{ñ}$-收缩和 $\mathbb{Ë}$-收缩。在每个分解定理中，下面的希尔伯特空间都允许由最后一个分量P提供的唯一正交分解。在确定组织结构中的核心作用${\mathrm{\Gamma }}_{ñ}$-收缩或 $\mathbb{Ë}$-收缩是由某些操作员铅笔的正性和与操作员相关联的唯一操作员元组的存在而产生的。 ${\mathrm{\Gamma }}_{ñ}$-收缩或 $\mathbb{Ë}$-收缩。