| dc.description.abstract | 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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