相似於不可約的算子

Abstract

本論文旨在探討在複希爾伯特空間上的算子，可相似於不可約算子之充份必要條件。 我們的成果如下: 在有限維的希爾伯特空間上，我們證明(1)可相似於不可約矩陣的 2x2 矩陣必不是純量矩陣；且反之亦然。(2)可相似於不可約矩陣的 nxn (n > 2) 矩陣 T 必不為 2 次矩陣，且對任意複數 a， T-aI 的秩不小 n/2；反之亦然。 在無窮維的希爾伯特空間上，針對加權(單側或雙側)移位算子或擬正規算子，我們有類似的結果。亦即，可相似於不可約算子的加權(單側或雙側)移位算子或擬正規算子 T 必不為2次算子，且對任意複數 a， T-aI 不為有限秩；反之亦然。此結果和有限維之矩陣，及正規算子(方資求, 蔣春瀾之研究結果)的情況吻合。 最後,我們針對C_0收縮算子加以探討。利用約當模型，得證可擬相似於不可約算子的C_0收縮算子T必不是2次算子，且對任意複數 a，T-aI 不為有限秩；反之亦然 。

In this thesis, we consider Hilbert space operators which are similar to irreducible ones. In the finite-dimentional case, we obtain the necessary and sufficient conditions for a complex square matrix $T$ to be similar to an irreducible one. We prove that (1) a $2\times 2$ matrix $T$ is similar to an irreducible matrix if and only if $T$ is not a scalar, and (2) an $n\times n$ ($n\geq 3$) matrix $T$ is similar to an irreducible matrix if and only if $T$ is not quadratic and rank $(T-\lambda I)\geq \dis\frac{n}{2}$ for any complex number $\lambda$. In the infinite-dimentional case, we prove an analogous result for weighted (unilateral or bilateral) shifts and quasinormal operators. Indeed, a weighted (unilateral or bilateral) shift or a quasinormal operator $T$ is similar to an irreducible operator if and only if $T$ is not quadratic and $T-\lambda I$ is not finite-rank for any complex number $\lambda$. It is obvious that the result is compatible with the ones in the finite-dimensional case and also with the work of C.K. Fong and C.L. Jiang on normal operators [8]. Finally, we consider $C_0$ contractions. Using the Jordan model of such operators, we prove that a $C_0$ contraction $T$ is quasisimilar to an irreducible operator if and only if $T$ is not quadratic and $T-\lambda I$ is not finite-rank for any complex number $\lambda$.