張量積的數值域

Abstract

在此一研究計畫中, 我們將探討, 對一個在複希伯特空間上的線性 有界算子A, 何時A 與A 的張量積的數值半徑會和A 的範數和A 的數 值半徑的乘積相等. 已知的是前者一定小於或等於後者. 我們希望 將二者的相等和A 的結構性質拉上關係, 並已獲致一些部分結果. 例 如, 現在已知, 設一個有限矩陣A 的範數為1, 則由此相等可推論出 A 有酉部分或A 的數值域是一個中心點為原點的圓盤. 設A 是一個非 負矩陣, 且其實數部分是不可約的, 則我們可以得到一個完整的刻 劃.

In this project, we want to study, for a bounded linear operator A on a complex Hilbert space, when the equality of the numerical radius of the tensor product of A with A and the product of the norm of A and the numerical radius of A occurs. It is known that the former is always less than or equal to the latter. We try to relate the equality to the structure properties of the operator A and have already obtained some partial results. For example, it is now known that, under the assumption of the norm of the finite matrix A equal to 1, the equality implies that either A has a unitary part or the numerical range of A is a circular disc centered at the origin. When A is an (entry-wise) nonnegative matrix with its real part (permutationally) irreducible, we do have a complete characterization.