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博碩士論文 etd-0709107-162359 詳細資訊
Title page for etd-0709107-162359
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
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
35
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
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
統計
Statistics
本論文已被瀏覽 5703 次,被下載 1348
The thesis/dissertation has been browsed 5703 times, has been downloaded 1348 times.
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.
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[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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