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Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem
Banach Journal of Mathematical Analysis ( IF 1.1 ) Pub Date : 2020-10-14 , DOI: 10.1007/s43037-020-00084-9
Santhosh Kumar Pamula

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $\delta >0$ there exist a compact operator $K$ with $\|K\|< \delta$, a partial isometry $W$ and a strongly irreducible operator $S$ on $\mathcal{H}$ such that \begin{equation*} T = (W+K) S. \end{equation*} We illustrate our result with an example. We also prove a quaternionic version of the Riesz decomposition theorem and as a consequence, show that if the spherical spectrum of a bounded quaternionic operator (need not be normal) is disconnected by a pair of disjoint axially symmetric closed subsets, then it is strongly reducible.

中文翻译:

四元数算符的强不可约分解和 Riesz 分解定理

令 $T$ 是右四元数希尔伯特空间 $\mathcal{H}$ 上的有界四元数正规算子。我们证明 $T$ 可以在强不可约的意义上被分解,也就是说,对于任何 $\delta >0$ 都存在一个紧算子 $K$,其中 $\|K\|< \delta$,一个部分等距 $ W$ 和 $\mathcal{H}$ 上的强不可约算子 $S$ 使得 \begin{equation*} T = (W+K) S. \end{equation*} 我们用一个例子来说明我们的结果。我们还证明了 Riesz 分解定理的四元数版本,因此,表明如果有界四元数算子(不必是正态的)的球谱被一对不相交的轴对称闭合子集断开,那么它是强可约的.
更新日期:2020-10-14
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