Browsing by Author "Tsai, Ming Cheng"

Now showing 1 - 4 of 4

Numerical radius inequality for C(0) contractions
(2009-03-01)
Wu, Pei Yuan
;
Gau, Hwa-Long
;
Tsai, Ming Cheng
;
應用數學系
;
Department of Applied Mathematics
We show that if A is a C(0) contraction with minimal function phi such that w(A) = w(S(phi)), where w(.) denotes the numerical radius of an operator and S(phi) is the compression of the shift on H(2 circle minus phi)H(2), and B commutes with A, then w(AB) <= w(A)parallel to B parallel to. This is in contrast to the known fact that if A = S(phi) (even on dimensional space) and B commutes with A, then w(AB) <= w parallel to A parallel to w(B) is not necessarily true. As a a finite consequence, we have w(AB) <= w(A)parallel to B parallel to for any quadratic operatorA and any B commuting with A. (c) 2007 Elsevier Inc. All rights reserved.
Numerical radius inequality for C-0 contractions
(2009-03-01)
Wu, Pei Yuan
;
Gau, Hwa-Long
;
Tsai, Ming Cheng
;
應用數學系
;
Department of Applied Mathematics
We show that if A is a C-0 contraction with minimal function phi such that w(A) = w(S(phi)), where w(.) denotes the numerical radius of an operator and S(phi) is the compression of the shift on (HH2)-H-2 circle minus phi, and B commutes with A, then w(AB) <= w(A)parallel to B parallel to. This is in contrast to the known fact that if A = S(phi) (even on dimensional space) and B commutes with A, then w(AB) <= w parallel to A parallel to w(B) is not necessarily true. As a a finite consequence, we have w(AB) <= w(A)parallel to B parallel to for any quadratic operatorA and any B commuting with A. (c) 2007 Elsevier Inc. All rights reserved.
Numerical ranges of weighted shift matrices
(2011-07-15)
Tsai, Ming Cheng
;
Wu, Pei Yuan
;
應用數學系
;
Department of Applied Mathematics
We show that if A is an n-by-n (n >= 3) matrix of the form (sic) then the boundary of its numerical range contains a line segment if and only if the a(j)'s are nonzero and the numerical ranges of the (n - 1)-by-(n - 1) principal submatrices of A are all equal. For n = 3, this is the case if and only if vertical bar a(1)vertical bar = vertical bar a(2)vertical bar = vertical bar a(3)vertical bar not equal 0, in which case W (A), the numerical range of A, is the equilateral triangular region with vertices the three cubic roots of aia2a3. For n = 4, the condition becomes vertical bar a(1)vertical bar = vertical bar a(3)vertical bar not equal 0 and vertical bar a(2)vertical bar = vertical bar a(4)vertical bar not equal 0, in which case W (A) is the convex hull of two (degenerate or otherwise) ellipses. (C) 2011 Elsevier Inc. All rights reserved.
Numerical ranges of weighted shift matrices with periodic weights
(2011-11-01)
Tsai, Ming Cheng
;
應用數學系
;
Department of Applied Mathematics
Let A be an n-by-n (n >= 2) matrix of the form [0 a(1) 0 a(n-1) a(n) 0] We show that if the a(j)'s are nonzero and their moduli are periodic, then the boundary of its numerical range contains a line segment. We also prove that partial derivative W (A) contains a noncircular elliptic arc if and only if the a(j)'s are nonzero, n is even, vertical bar a(1)vertical bar = vertical bar a(3)vertical bar = ... = vertical bar a(n-1)vertical bar, vertical bar a(2)vertical bar = vertical bar a(4)vertical bar = ... = vertical bar a(n)vertical bar and vertical bar a(1)vertical bar not equal vertical bar a(2)vertical bar. Finally, we give a criterion for A to be reducible and completely characterize the numerical ranges of such matrices. (C) 2011 Elsevier Inc. All rights reserved.