矩陣和算子數值域的研究

Abstract

在這項三年研究計畫中，我們將深入探討有限矩陣或希伯特空間上有 界線性算子的數值域的性質。延續我們過去在這一題材上的研究工作，我 們計畫探究下列幾個題目： (一)在第一年，我們計畫研究兩類特殊形式的矩陣和算子：[ ] i, j=1 n ij a , 其 中當(i, j) ≠ (1,2), (2,3),..., (n −1, n), (n,1) 時，皆有0 ij a = ，和, 0 [ ] ij i j a ∞ = ，其中j ≠ i + 1 時，皆有0 ij a = 。我們主要考慮前者的數值域的邊界上何時存在一線段及 後者數值域何時為一開圓盤。 (二)在第二年，我們將研究一個算子或矩陣A 的Crawford 數c(A)和 C(A)，即由原點到A 的數值域或其邊界的距離。我們希望如同古典的結果 1/ lim ( ) k k kA =rA (此處r(A)表示A 的特徵值絕對值最大者) 般得到 limsup ( k)1/k k c A 和limsup ( k)1/k kC A 和A 的特徵值或譜之間的關係。 (三)在第三年，我們希望將過去所作關於一個矩陣的所有高秩數值域 如何決定此矩陣的結果作一總結。另一方面則想推廣Halmos 關於收縮算 子酉膨脹和其數值域的關係到無窮維空間上收縮算子的高秩數值域上。

In this three-year project, we plan to initiate a detailed study of properties of numerical ranges of finite square matrices and bounded linear operators on a Hilbert space. Based on our previous works, we intend to investigate the following topics concerning the numerical ranges in the coming three years: (1) In the first year, we will determine the characteristics of the numerical ranges of matrices and operators of certain special forms such as A = [ ] i, j=1 n ij a with ij a = 0 for all i and j with (i, j) ≠ (1,2), (2,3),..., (n − 1, n), (n,1) , and B = [ ] , =0 ∞ ij i j a with ij a = 0 for all i and j with j ≠ i + 1. The main concern will be the existence of line segments on the boundary of the numerical range for the matrix A and the closedness or openness of the numerical range for the operator B. (2) For the second year, we plan to concentrate on the study of the Crawford number c(A) (resp., generalized Crawford number C(A)) of a matrix or an operator A. These are defined as the distance from the origin to the (resp., boundary of the) numerical range W(A) of A. Following a recent paper of M.-D. Choi and C.-K. Li, we will try to determine the limit suprema of c(Ak )1/ k and C(Ak )1/ k as k approaches infinity. These give the asymptotic distances of the origin from the numerical ranges of powers of A and are analogous to classical results for norms and numerical radii. From our preliminary observations, they can be expressed in terms of the eigenvalues or the spectrum of A. (3) In the third year, we will try to complete our previous works on higher-rank numerical ranges. Recall that for a matrix or an operator A and a positive integer k, the rank-k numerical range is, be definition, the subset A z C PAP zP k Λ() = { ∈ : = for some rank-k orthogonal projection P}. Note that ( ) 1 Λ A is just the classical numerical range W(A). Properties of such higher-rank numerical ranges have been pursued feverishly in recent years by researchers. One of the questions we have been pondering is to what extent the rank-k numerical ranges of A determine the matrix A. Another result we haven’t been able to completely solve is whether a contraction A on H with its defect indices d rank(I A* A) A ≡ − and ( *) * d rank I AA A ≡ − equal to each other, say, to d has the closure of its rank-k numerical range equal to the intersection of the rank-k numerical ranges of its unitary dilations on spaceH ⊕ Cd . The case when k = 1 and no restriction is imposed on the space on which the unitary dilations of A act was solved by M.-D. Choi and C.-K. Li (2001). Our assertion for A on a finite-dimensional space is also true by H.-L. Gau, C.-K. Li and P. Y. Wu (2010). The generalization to the infinite-dimensional case seems more difficult.