在第一章中，主要討論的是加權移位矩陣的各種特性。首先，我們提出兩個加權位移矩陣互為正交基底變換的等價條件。接下來，我們將就加權移位矩陣的可分解性來探討，我們也發現了它可分解時的等價條件。最後，我們將探討加權移位矩陣的數值域。這邊又分成兩個部分，第一部份是討論兩個加權移位矩陣何時他們的數值域會相同； 第二部分是討論加權移位矩陣的數值域邊界何時會出現直線段。
在第二章中，我們將在 Banach 空間下討論著名的三角不等式的優化和當函數本身是強可積函數時其反向不等式。我們還討論了當函數為Lp函數時第二類廣義三角不等式的優化 。我們也針對這兩種情況下，不等式號成立的條件 。

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.

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