| 摘要(英) |
Assume that
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.
|
| 參考文獻 |
[1] S. S. Dragomir, Reverses of the triangle inequality in Banach spaces, J. Inequal.
Pure and Appl. Math. 6, (5) (2005), Art. 129, pp. 46.
[2] N. Dunford and J. T. Schwartz, Linear Operators, Part 1, Interscience Publishers,
Inc., New York, 1957.
[3] H.-L. Gau and P. Y. Wu, Companion matrices: reducibility, numerical ranges
and similarity to contractions, Linnear Algebra Appl. 383 (2004) 127{142
[4] H.-L. Gau and P. Y.Wu, Numerical ranges of nilpotent operators, Linear Algebra
Appl. 429 (2008) 716{726.
[5] K. Gustafson and D. K. M. Rao, Numerical Range. The Field of Values of Linear
Operators and Matrices, Springer, New York, 1997.
[6] P. R. Halmos, A Hilbert Space Problem Book, second ed., Springer, New York,
1982.
[7] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cam-
bridge, 1985.
[8] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ.
Press, Cambridge, 1991.
[9] 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.
[10] 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).
[11] 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.
[12] J. M. Rassias, Solutions of the Ulam stability problem for Euler-Langrage
quadratic mappings. J. Math. Anal. 220 (1998), 613-639.
[13] H. L. Royden, Real Analysis, 3rd ed., Prentice Hall, New Jersey, 1989.
[14] Q. F. Stout, The numerical range of a weighted shift, Proc. Amer. Math. Soc.
88 (1983) 495{502
[15] S. Saitoh, Generalizations of the triangle inequality, J. Inequal. Pure and Appl.
Math. 4, (3) (2003), Art. 62, pp. 5.
[16] Sin-Ei Takahasi, J. M. Rassias, S. Saitoh, and Y. Takahashi, Re¯ned generaliza-
tions of the triangle inequality on Banach space, preprint.
[17] M.-C. Tsai, Numerical ranges of weighted shift matrices with periodic weights,
Linear Algebra Appl. 435 (2011) 2296{2302.
[18] M.-C. Tsai and P. Y. Wu, Numerical ranges of weighted shift matrices, Linear
Algebra Appl. 435 (2011) 243{254.
[19] C.-Y. Hus, Re¯nements of triangle inequality and Jensen’’s inequality, Matser
dissertation, National Center University, 2007.
|
[Endnote RIS 格式]
[相關文章]
[文章引用]
[完整記錄]
[館藏目錄]
[檢視]
plurk
funp
live
udn
HD
myshare
netvibes
friend
youpush
delicious
baidu
Google bookmarks
del.icio.us
hemidemi
facebook
twitter
google
reddit