有限矩陣及有界算子數值域之研究(I)

Abstract

婓森珨爛旃噶??笢ㄛ扂?旃噶洷皎杻諾嶲奻腔?俶呾赽睿衄癹撻?腔 在此一三年研究計劃中，我們將研究希伯特空間上的線性算子和有限矩陣的 數值域的性質。主要將探討四個不同的問題： (1) 一個n 階矩陣數值域的邊界上最多可有幾條線段？ (2) 給定n+1 個單位圓上不同的點1 a ,…, 1 n a 及[n/2]個由1 a ,…, 1 n a 所產生 的凸包內的點1 b ,…, ] 2 / [n b , 則是否存在唯一的n S -矩陣A，使得數值域 W(A)和多邊形1 a ,…, 1 n a 的n+1 個邊皆相切，且1 b ,…, ] 2 / [n b 為A 的特徵 值？ (3) 設A 和B 是同一希伯特空間上的可換算子，且A 是一二次型算子，則 B A w AB w ) ( ) ( 和) ( ) ( B w A AB w 是否成立？ (4) 設A 是一緊緻算子，其數值域W(A)係包含於單位圓盤D 內的閉集合，且 W(A)和單位圓相交於無窮多點，則是否D A WIn this three-year project, we plan to study certain properties of the numerical ranges of bounded linear operators on a Hilbert space and finite matrices. More specifically, we will consider the following four questions: (1) What is the maximum number of line segments on the boundary of the numerical range of an n-by-n matrixˋ (2) Does there exist a unique n S -matrix with half of its eigenvalues and one of its (n+1)-by-(n+1) unitary dilation givenˋ (3) If A and B are commuting operators on a Hilbert space and A is quadratic, then do B A w AB w ) ( ) ( ≒ and ) ( ) ( B w A AB w ≒ holdˋ (4) If A is a compact operator with its closed numerical range W(A) contained in the closed unit disc D and intersecting the unit circle at infinitely many points, then must W(A) equal