摘要:

這篇論文討論兩個問題：

(i).
研究作用在矩陣代數上的一類線性映射，它們在一個小集合上保持譜。

(ii).
我們應用 Choi-Kraus 定理，以研究正規矩陣 (算子) 之間的完全正插值問題。我們給出其存在性的充份必要條件，這些條件依賴於
數值域的包含關係和矩陣擴張問題。

In this thesis, we will consider the following problems:

(i).
We study linear maps of matrix algebras, which preserve some spectral functions on a small
subset of mn x mn matrices.

(ii).
We study completely positive interpolations between normal matrices (operators) using
their spectrum and the Choi-Kraus form. We obtain a necessary and sufficient condition
for the existence of a completely positive interpolation in terms of conditions about
numerical ranges and dilations.

Keywords:

目 錄
論文審定書…………………………………………………………… i
論文誌謝……………………………………………………………… ii
中文摘要………………………………………………………….….. iii
Abstract ………………………………………..……………………. iv

Contents
Chapter 1: Introduction 1
1.1 Linear maps preserving spectral properties of tensor products of matrices . . 1
1.2 Completely positive interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2: Preliminaries 7
2.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Banach spaces and Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Linear operators on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Spectra of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Numerical ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.8 Compact and Schatten-p class operators . . . . . . . . . . . . . . . . . . . . 16
2.9 A technical lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 3: Linear maps preserving spectral properties of tensor product
of low rank matrices 20
3.1 Spectrum and numerical range preservers . . . . . . . . . . . . . . . . . . . 20
3.2 Spectral radius preservers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Numerical radius preservers . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 4: Completely Positive Interpolation 34
4.1 Interpolations and numerical range inclusions . . . . . . . . . . . . . . . . . 34
4.2 Completely positive interpolations preserving approximate units or trace . . . 47
4.3 Completely positive interpolations between commutative families . . . . . . 53
4.4 Extension to the General Case . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Computational considerations . . . . . . . . . . . . . . . . . . . . . . . . . 64
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