  博碩士論文 942401005 詳細資訊

 以作者查詢圖書館館藏 、以作者查詢臺灣博碩士 、以作者查詢全國書目 、勘誤回報 、線上人數：47 、訪客IP：3.228.21.186

(Weighted shift matrices and refinements of generalized triangle inequalities.)

 ★ 關於超循環算子的一些基本性質 ★ r維近似算子的收斂速度 ★ 摺合型積分方程之收斂性,可微性與可容許空間的研究。 ★ 某些正線性算子作用在無界連續函數上的估計 ★ 橢圓形數值域之四階方陣 ★ Daugavet方程式在算子序列與 算子函數下的推廣 ★ 數值域邊界上之線段 ★ 正規壓縮算子與正規延拓算子 ★ 論Lyons不等式和關於Meyer-König-Zeller近似算子的估計 ★ 加權排列矩陣及加權位移矩陣之數值域 ★ 可分解友矩陣之數值域 ★ 可分解友矩陣之研究 ★ 關於函數及積分,演化方程式解之行為的探討 ★ 關於巴氏空間上連續函數的近乎收斂性 ★ 三角不等式與Jensen不等式之精化 ★ 缺陷指數為1的矩陣之研究

1. 本電子論文使用權限為同意立即開放。
2. 已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
3. 請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

all aj’’s are nonzero and B is a n-by-n weighted shift matrix with weights bj ’’s. We
show that B is unitarily equivalent to A if and only if a1 ￠ ￠ ￠ an = b1 ￠ ￠ ￠ bn and,
for some ¯xed k, 1 · k · n, jbj j = jak+j j (an+j ’ aj) for all j. Next, we show
that A is reducible if and only if A has periodic weights, that is, for some ¯xed k,
1 · k · bn=2c, n is divisible by k, and jaj j = jak+j j for all 1 · j · n!k. Finally, we
prove that A and B have the same numerical range if and only if a1 ￠ ￠ ￠ an = b1 ￠ ￠ ￠ bn
and Sr(ja1j2; : : : ; janj2) = Sr(jb1j2; : : : ; jbnj2) for all 1 · r · bn=2c, where Sr’’s are
the circularly symmetric functions. Let A[j] denote the (n ! 1)-by-(n ! 1) principal
submatrix of A obtained by deleting its jth row and jth column. We show that the
boundary of numerical range W(A) has a line segment if and only if the aj’’s are
nonzero and W(A[k]) = W(A[l]) = W(A[m]) for some 1 · k < l < m · n. This
re¯nes previous results which Tsai andWu made on numerical ranges of weighted shift
matrices. In Chapter 2, we discuss re¯nements of the well-known triangle inequality
and it’’s reverse inequality for strongly integrable functions with values in a Banach
space X. We also discuss re¯nement for the Lp functions in the second kind of
generalized triangle inequality . For both cases, the attainability of the equality is
also investigated.

★ 廣義三角不等式

★ Weighted shift
★ Numerical range

Chapter 1. Weighted shift matrices
1. Introduction 1
2. Unitary equivalence 5
3. Reducibility 9
4. Numerical ranges 14
Chapter 2. Refinements of generalized triangle inequalities
1. Introduction 37
2. Sharp triangle inequality and its reverse for integrable functions 39
3. Specializations to series 52
4. Generalization of the triangle inequality of the second kind 55
References 59

Pure and Appl. Math. 6, (5) (2005), Art. 129, pp. 46.
 N. Dunford and J. T. Schwartz, Linear Operators, Part 1, Interscience Publishers,
Inc., New York, 1957.
 H.-L. Gau and P. Y. Wu, Companion matrices: reducibility, numerical ranges
and similarity to contractions, Linnear Algebra Appl. 383 (2004) 127{142
 H.-L. Gau and P. Y.Wu, Numerical ranges of nilpotent operators, Linear Algebra
Appl. 429 (2008) 716{726.
 K. Gustafson and D. K. M. Rao, Numerical Range. The Field of Values of Linear
Operators and Matrices, Springer, New York, 1997.
 P. R. Halmos, A Hilbert Space Problem Book, second ed., Springer, New York,
1982.
 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cam-
bridge, 1985.
 R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ.
Press, Cambridge, 1991.
 M. Kato, K.-S. Saito, and T. Tamura, Sharp triangle inequality and it’’s reverse
in Banach space, Math. Inequal. Appl., 10 (2007), 451-460.
 R. Kippenhahn, AUber den Wertevorrat einer Matrix, Math. Nachr. 6 (1951) 193{
228 (English translation: P. F. Zachin and M. E. Hochstenbach, On the numerical
range of a matrix, Linear and Multilinear Algebra 56 (2008) 185{225).
 D. S. Mitrinovi¶c, J. J. Pecari¶c and A. M. Fink, Classical and New Inequalities
in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993.
 J. M. Rassias, Solutions of the Ulam stability problem for Euler-Langrage
quadratic mappings. J. Math. Anal. 220 (1998), 613-639.
 H. L. Royden, Real Analysis, 3rd ed., Prentice Hall, New Jersey, 1989.
 Q. F. Stout, The numerical range of a weighted shift, Proc. Amer. Math. Soc.
88 (1983) 495{502
 S. Saitoh, Generalizations of the triangle inequality, J. Inequal. Pure and Appl.
Math. 4, (3) (2003), Art. 62, pp. 5.
 Sin-Ei Takahasi, J. M. Rassias, S. Saitoh, and Y. Takahashi, Re¯ned generaliza-
tions of the triangle inequality on Banach space, preprint.
 M.-C. Tsai, Numerical ranges of weighted shift matrices with periodic weights,
Linear Algebra Appl. 435 (2011) 2296{2302.
 M.-C. Tsai and P. Y. Wu, Numerical ranges of weighted shift matrices, Linear
Algebra Appl. 435 (2011) 243{254.
 C.-Y. Hus, Re¯nements of triangle inequality and Jensen’’s inequality, Matser
dissertation, National Center University, 2007.

(Sen-Yen Shaw、Hwa-Long Gau)