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姓名 張承鈞(Cheng-chun Chang) 查詢紙本館藏 畢業系所 數學系 論文名稱 位移算子其有限維壓縮算子的反矩陣
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摘要(中) 論文名稱：位移算子其有限維壓縮算子的反矩陣
頁數：27 頁
校所組別：國立中央大學數學所甲組
研究生：張承鈞 指導教授：高華隆
論文提要內容：
令 A 為一個 n 階矩陣，其缺陷指數 d_A 為 rank (I_n − A∗A)。
本論文探討關於「缺陷指數為 1 的矩陣」其性質之刻劃。
令 S_n ≡ { A ∈ M_n : d_A = 1 and |λ| < 1 for all λ ∈ σ(A) } 和 S_n^−1 ≡ { A ∈ M_n : d_A = 1 and |λ| > 1 for all λ ∈ σ(A)}。我們針對「缺陷指數為 1 的矩陣」研究其極分解、數值域、範數和冪次對缺陷指數的影響。進一步而言，我們證明了 S_n^−1-矩陣其實部的特徵值皆無重根。此外，我們也對 S_n^−1-矩陣的數值域做了詳細的刻劃。最後我們給出任一矩陣為S_n−1-矩陣的充分必要條件。
摘要(英) Let M_n be the algebra of all n-by-n complex matrices. Let A be an n-by-n matrix. The defect index of A is defined and denoted by d_A ≡ rank (I_n − A∗A). In this thesis, we study some unitary-equivalence properties of matrices with defect index one. We denote S_n ≡ {A ∈ M_n : d_A = 1 and |λ| < 1 for all λ ∈ σ(A)} and S_n^−1 ≡ {A ∈ M_n : d_A = 1 and |λ| > 1 for all λ ∈ σ(A)}. We want to give some characterizations of the polar decompositions, numerical ranges, norms and defect indices of powers of matrices with defect index one. In particular, we show that the eigenvalues of the real part of operators in S_n^−1 are simple. Next, we give some characterizations of the numerical ranges of S_n^−1-matrices. Finally, we find the sufficient and necessary conditions for a matrix in the class S_n^−1.
關鍵字(中) ★ 矩陣
★ 數值域
★ 缺陷指數關鍵字(英) ★ Norm
★ Numerical range
★ Defect index
★ Eigenvalue論文目次 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Basic Properties for Numerical Ranges . . . . . . . . . . . . . 3
2.2 Powers of a Contraction . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Classifications of Matrices with Defect Index One. . . . . . 6
2.4 Matrices of Classes S_n and S_n^−1 . . . . . . . . . . . . . . 7
3 Characterizations of S_n^−1-matrices . . . . . . . . . . . . . . . 10
3.1 Spectrum of Real Part . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Numerical Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Defect Indices of Powers . . . . . . . . . . . . . . . . . . . . . . 19
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
參考文獻 [1] H. Bercovici, Operator Theory and Arithmetic in H∞, Amer. Math. Soc., Providence, 1988.
[2] R. Bhatia, Matrix Analysis, Springer, New York, 1997.
[3] H.-L. Gau, Numerical ranges of reducible companion matrices, Linear Alge-bra Appl., 432 (2010), 1310-1321.
[4] H.-L. Gau and P. Y. Wu, Numerical Range of S( ), Linear and Multilinear Algebra, 45 (1998), 49-73.
[5] H.-L. Gau and P. Y. Wu, Lucas’ Theorem Refined, Linear and Multilinear Algebra, 45 (1999), 359-373.
[6] H.-L. Gau and P. Y. Wu, Finite Blaschke products of contractions, Linear Algebra Appl., 368 (2003), 359-370.
[7] H.-L. Gau and P. Y. Wu, Defect indices of powers of a contraction, Linear Algebra Appl., 432 (2010), 2824-2833.
[8] K. E. Gustafson and D. K. M. Rao, Numerical Range: the Field of Values of Linear Operators and Matrices, Springer, New York, 1997.
[9] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, 1985.
[10] C.-K. Li, A note on the unitary part of a contraction, preprint.
[11] B. Sz.-Nagy and C. Foia﹐s, Harmonic Analysis of Hilbert Space Operators, North Holland, Amsterdam, 1970.
[12] P. Y. Wu, Polar Decompositions of C0(N) contractions, Integral Equations Operator Theory, 56(2006), 559-569.
[13] P. Y. Wu, Numerical Ranges of Hilbert Space Operators, preprint.
[14] S.-C. Wu, A study on Matrices of Defect Index One, National Central Univ., Taiwan, June 2008.
指導教授 高華隆(Hwa-long Gau) 審核日期 2011-5-3 推文 plurk
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