有界算子數值域的研究

Abstract

在此一三年期的研究計畫中，我們將對在希伯特空間上的有界線性算子的數值域 作一有系統而詳盡的研究。第一年中，我們將集中心力於有限維空間上的算子或有限 矩陣。基於過去所作的相關研究成果，我們將專注於兩類矩陣，即Sn -矩陣和伴隨矩陣。 第二年將考慮其他類型的矩陣，例如Toeplitz 矩陣、Hankel 矩陣和形如, 1 [ ]n ij i j a = 之矩陣， 其中對每一k (1 ? k ? n)，i=k或j=k時， ij a 之值均相同。我們希望對比此等矩陣之數 值域和其無窮維空間上的對應算子二者性質之共同與相異之處。第三年將專注於無窮 維空間上算子數值域之研究。我們特別想解決一已懸疑多年的問題：設A 為二次型算 子且B和A可交換，則是否w(AB) ? A w(B)和w(AB) ? w(A) B 成立，此處w(?)表示 算子之數值半徑。

In this three-year project, we plan to launch a detailed study of properties of the numerical ranges of bounded linear operators on a complex Hilbert space. For the first year, we will concentrate on operators on finite-dimensional spaces, which amount to the same as finite matrices. Based on our previous works, we will concentrate on two types of matrices, namely, the n S -matrices and companion matrices. In the second year, we move onto other types of finite matrices such as the Toeplitz matrices, Hankel matrices and matrices of the form , 1 [ ]n ij a i j= with the aij 刪s identical for i=k or j=k (1 . k . n) . Properties of their numerical ranges will be contrasted with the ones for their infinite-dimensional analogues. The third year will be devoted to the study of numerical ranges of infinite-dimensional operators. We have in mind in particular of proving the numerical radius inequalities w(AB) . A w(B) and w(AB) . w(A) B for a quadratic operator A commuting with an operator B .