友矩陣和規範算子的數值域

Abstract

在本篇論文中，我們研究兩類算子的數值域。針對友矩陣部分，我們證明 可分解友矩陣的數值域會包含相對應的 友矩陣的數值域的充要調件為 可分解友矩陣的行列式值的絕對值會大於1。然而，相對應的結果在一般的 可分友矩陣並不正確。針對規範算子的部份，我們將數值域以乘積算子的表現函數來表示。

In this thesis, we study the numerical ranges of two kinds of operators. For companion matrices, we show that the numerical range of a 3-by-3 reducible companion matrix C(p) contains the numerical range of the 2-by-2 companion matrix C((1=3)p0) if and only if the absolute value of its determinant is greater than 1. However, the corresponding assertion for n-by-n reducible companion matrices is false. For a normal operator, we express its numerical range in terms of the function in its multiplication operator representation.