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論文名稱 Title |
兩個二次冪等元與兩個二次冪零算子的線性組合之可逆性 On the invertibility of linear sums of two idempotents and of two square zero operators |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
35 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2007-06-15 |
繳交日期 Date of Submission |
2007-07-09 |
關鍵字 Keywords |
二次冪零算子、部分等距、二次冪等元、可逆性、n次冪等、n次冪零 partial isometry, idempotent, n-potent, invertibility, nilpotent, square-zero operator |
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統計 Statistics |
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中文摘要 |
我們回顧了兩個二次冪等元的線性組合之可逆性,若P與Q皆為二次冪等元,對於所有的非零係數a與b使得a+b非零,則aP+bQ的可逆性與P+Q的可逆性是等價的。在本論文中,我們考慮了當P與Q為二次冪零算子、部分等距、n次冪等還有n次冪零的情形,其中我們證明了在P跟Q為二次冪零算子的情形,與二次冪等元同樣的結果是會成立的,同時我們也給予了一些反例說明了在P跟Q為其它情形時,同樣的結果在一般狀況下不會成立。 |
Abstract |
Let P and Q be two idempotents, we review the results about the equivalence between the invertibility of a linear combination aP +bQ and that of P +Q, where a and b are any nonzero complex numbers with a + b eq 0. It is possible to extend the results to the case P and Q are square-zero elements. However, we will show that these extensions are impossible in general for P and Q being partial isometries or n-potents with n geq 3. We will show in case P and Q are square-zero elements, the invertibility of P +Q is equivalent to that of aP +bQ for nonzero a, b. |
目次 Table of Contents |
1 Review of known results 2 2 New results 21 |
參考文獻 References |
[1] J.K. Baksalary and O.M. Baksalary, Nonsigularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388:25-29 (2004). [2] J.B. Conway, A Course in Functional Analysis, 2nd ed., Springer, New York, 1990. [3] H. Du, X. Yao, and C. Deng, Invertibility of linear combinations of two idempotents, Proc. Amer. Math. Soc. 134:1451-1457 (2006). [4] H. Du, C. Deng, M. Mbekhta and V. M¨uller, On spectral properties of linear combinations of idempotents, preprint. [5] P.A. Fillmore, On sums of projection, J. Funct. Anal. 4:146-152 (1969). [6] P.A. Fillmore, On similarity and the diagonal of a matrix, Amer. Math. Monthly 76:167-169 (1969). [7] H.L. Gau and P.Y. Wu, Fredholmness of linear combinations of two idempotents, preprint. [8] J. Groß, G. Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21:390-395 (1999). [9] J.J. Koliha and V. Rakoˇcevi´c, The nullity and rank of linear combinations of idempotent matrices, Linear Algebra Appl. 418:11-14 (2006). [10] J.J. Koliha and V. Rakoˇcevi´c, Stability Theorems for Linear Combinations of Idempotents, Integr. equ. oper. theory Online First (2007). [11] C. Pearcy and D. Topping, Sums of small numbers of idempotents, Michigan Math. J. 14:453-465 (1967). [12] J.H. Wang and P.Y Wu, Sums of square-zero operators, Studia Math. no. 2, 99:115-127 (1991). [13] P.Y. Wu, Sums of idempotent matrices, Linear Algebra Appl. 142:43-54 (1990). |
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