Browsing by Author "Wang, Kuo-Zhong"

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Circular numerical ranges of partial isometries
(2016-01-02)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
Let A be an n-by-n partial isometry whose numerical range W(A) is a circular disc with centre c and radius r. In this paper, we are concerned with the possible values of c and r. We show that c must be 0 if n is at most 4 and conjecture that the same is true for the general n. As for the radius, we show that if W(A) = {z is an element of C : vertical bar z vertical bar <= r}, then the set of all possible values of r is {0, cos(pi/(n + 1))}boolean OR [cos(pi/3), cos(pi/n)]. Indeed, it is shown more precisely that for dim ker A = k, 1 <= k <= n, the possible values of r are those in the interval [cos(pi/inverted right perpendicularn/kinverted left perpendicular + 1)), cos(pi/(n - k + 2))]. In the proof process, we also characterize n-by-n partial isometries which are (unitarily) irreducible. The paper is concluded with a question on the rotational invariance of nilpotent partial isometries with circular numerical ranges centred at the origin.
CONSTANT NORMS AND NUMERICAL RADII OF MATRIX POWERS
(2019-12-01)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
For an n-by-n complex matrix A, we consider conditions on A for which the operator norms parallel to A(k)parallel to (resp., numerical radii w(A(k))), k >= 1, of powers of A are constant. Among other results, we show that the existence of a unit vector x in C-n satisfying vertical bar < A(k)x,x >vertical bar = w(A(k)) = w(A) for 1 <= k <= 4 is equivalent to the unitary similarity of A to a direct sum lambda B circle plus C, where vertical bar lambda vertical bar = 1, B is ideinpotent, and C satisfies w(C-k) <= w(B) for 1 <= k <= 4. This is no longer the case for the norm: there is a 3-by-3 matrix A with parallel to A(k)x parallel to = parallel to A(k)parallel to = root 2 for some unit vector x and for all k >= 1, but without any nontrivial direct summand. Nor is it true for constant numerical radii without a common attaining vector. If A is invertible, then the constancy of parallel to A(k)parallel to (resp., w(A(k))) for k = +/- 1, +/- 2, ... is equivalent to A being unitary. This is not true for invertible operators on an infinite-dimensional Hilbert space.
CRAWFORD NUMBERS OF COMPANION MATRICES
(2016-12)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
The (generalized) Crawford number C(A) of an n-by-n complex matrix A is, by definition, the distance from the origin to the boundary of the numerical range W(A) of A. If A is a companion matrix [GRAPHICS] then it is easily seen that C(A) >= cos(pi/n). The main purpose of this paper is to determine when the equality C(A) = cos(pi/n) holds. A sufficient condition for this is that the boundary of W(A) contains a point lambda for which the subspace of C-n spanned by the vectors x with < Ax, x > = lambda parallel to x parallel to(2) has dimension 2, while a necessary condition is Sigma(n-2)(j=0) a(n-j)e((n-j)i theta) sin ((j + 1)pi/n) = sin(pi/n) for some real theta. Examples are given showing that in general these conditions are not simultaneously necessary and sufficient. We then prove that they are if A is (unitarily) reducible. We also establish a lower bound for the numerical radius w(A) of A: w(A) >= cos(pi/(n+ 1)), and show that the equality holds if and only if A is equal to the n-by-n Jordan block.
CRAWFORD NUMBERS OF COMPANION MATRICES
(2016-12)
Gau, Hwa-Long
;
Wang, Kuo-Zhong
;
Wu, Pei Yuan
;
應用數學系
;
Department of Applied Mathematics
The (generalized) Crawford number C(A) of an n-by-n complex matrix A is, by definition, the distance from the origin to the boundary of the numerical range W(A) of A. If A is a companion matrix [GRAPHICS] then it is easily seen that C(A) >= cos(pi/n). The main purpose of this paper is to determine when the equality C(A) = cos(pi/n) holds. A sufficient condition for this is that the boundary of W(A) contains a point lambda for which the subspace of C-n spanned by the vectors x with < Ax, x > = lambda parallel to x parallel to(2) has dimension 2, while a necessary condition is Sigma(n-2)(j=0) a(n-j)e((n-j)i theta) sin ((j + 1)pi/n) = sin(pi/n) for some real theta. Examples are given showing that in general these conditions are not simultaneously necessary and sufficient. We then prove that they are if A is (unitarily) reducible. We also establish a lower bound for the numerical radius w(A) of A: w(A) >= cos(pi/(n+ 1)), and show that the equality holds if and only if A is equal to the n-by-n Jordan block.
Crawford numbers of powers of a matrix
(2010-12-30)
Wang, Kuo-Zhong
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Wu, Pei Yuan
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Gau, Hwa-Long
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應用數學系
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Department of Applied Mathematics
For an n-by-n matrix A, its Crawford number c(A) (resp., generalized Crawford number C(A)) is, by definition, the distance from the origin to its numerical range W(A) (resp., the boundary of its numerical range partial derivative W(A)). It is shown that if A has eigenvalues lambda(1), ..., lambda(n) An arranged so that vertical bar lambda(1)vertical bar >= ... >= vertical bar lambda(n)vertical bar, then (lim) over bar (k) c(A(k))(1/k) (resp., (lim) over bar (k) C(A(k))(1/k))equals 0 or vertical bar lambda(n)vertical bar (resp., vertical bar lambda(j)vertical bar for some j, 1 <= j <= n). For a normal A. more can be said, namely, (lim) over bar (k) c(A(k))(1/k) = vertical bar lambda(n)vertical bar (resp., (lim) over bar (k) C(A(k))(1/k) = vertical bar lambda(j)vertical bar for some j, 3 <= j <= n). In these cases, the above possible values can all be assumed by some A. (C) 2010 Elsevier Inc. All rights reserved.
Diagonals and numerical ranges of weighted shift matrices
(2013-01-01)
Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
For any n-by-n matrix A, we consider the maximum number k = k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary partial derivative W (A) of the numerical range W (A) of A. For any such compression, we give a standard model under unitary equivalence for A. This is then applied to determine the value of k(A) for A of size 3 in terms of the shape of W (A). When A is a matrix of the form (0 W-1 ... 0 ... w(n-1) w(n) 0 ), we show that k(A) = n if and only if either vertical bar w(1)vertical bar = ... = vertical bar w(n)vertical bar or n is even and vertical bar w(1)vertical bar = vertical bar w(3)vertical bar = ... = vertical bar w(n-1)vertical bar and vertical bar w(2)vertical bar = vertical bar w(4)vertical bar = ... = lwn For such matrices A with exactly one of the wi's zero, we show that any k, 2 <= k <= n - 1, can be realized as the value of k(A), and determine exactly when the equality k(A) = n - 1 holds. (C) 2012 Elsevier Inc. All rights reserved.
Equality of higher-rank numerical ranges of matrices
(2014-05-04)
Chang, Chi-Tung
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Gau, Hwa-Long
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Wang, Kuo-Zhong
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應用數學系
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Department of Applied Mathematics
Let denote the rank- numerical range of an -by- complex matrix . We give a characterization for , where , via the compressions and the principal submatrices of . As an application, the matrix satisfying , where is the classical numerical range of and , is under consideration. We show that if for some , then is unitarily similar to , where is a 2-by-2 matrix, is a -by- matrix and .
Equality of numerical ranges of matrix powers
(2019-10-01)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
For an n-by-n matrix A, we determine when the numerical ranges W(A(k)), k >= 1, of powers of A are all equal to each other. More precisely, we show that this is the case if and only if A is unitarily similar to a direct sum B circle plus C, where B is idempotent and C satisfies W(C-k) subset of W (B) for all k >= 1. We then consider, for each n >= 1, the smallest integer k(n) for which every n-by-n matrix A with W(A) = W(A(k)) for all k, 1 <= k <= k(n), has an idempotent direct summand. For each n >= 1, let p(n) be the largest prime less than or equal to n + 1. We show that (1) k(n) >= p(n) for all n, (2) if A is normal of size n, then W(A) = W(A(k)) for all k, 1 <= k <= p(n), implies A having an idempotent summand, and (3) k(1) = 2 and k(2) = k(3) = 3. These lead us to ask whether k(n) = p(n) holds for all n >= 1. (C) 2019 Elsevier Inc. All rights reserved.
Extremality of numerical radii of matrix commutators and Jordan products
(2017-11-15)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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應用數學系
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Department of Applied Mathematics
It is known that the numerical radius of the commutator (resp., Jordan product) AB - BA (resp., AB + BA) of two n-by-n matrices A and B is related to those of A and B by w(AB +/- BA) <= 4w(A)w(B). In this paper, we give complete characterizations of A and B for which the equality is attained. The matrices involved can be considered as elaborate generalizations of the equality-attaining A = [0 a (0 0)] and B = [0 0 (b) (0)]. (C) 2017 Elsevier Inc. All rights reserved.
Matrix powers with circular numerical range
(2020-10-01)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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應用數學系
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Department of Applied Mathematics
Let K-2 = [GRAPHICS}, K-n be the n x n weighted shift matrix with weights root 2, [GRAPHICS}, root 2 for all n >= 3, and K-infinity be the weighted shift operator with weights root 2, 1, 1, 1, .... In this paper, we show that if an n x n nonzero matrix A satisfies W(A(k)) = W(A) for all 1 <= k <= n, then W(A) cannot be a (nondegenerate) circular disc. Moreover, we also show that W(A) = W(A(n-1)) = {z is an element of C : vertical bar z vertical bar <= 1} if and only if A is unitarily similar to K-n. Finally, we prove that if T is a numerical contraction on an infinite-dimensional Hilbert space H, then lim(n ->infinity) parallel to T(n)x parallel to = root 2 for some unit vector x is an element of H if and only if T is unitarily similar to an operator of the form K-infinity circle plus T' with w(T') <= 1. (C) 2020 Elsevier Inc. All rights reserved.
Maximizing numerical radii of weighted shifts under weight permutations
(2012-10-15)
Chang, Chi-Tung
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Wang, Kuo-Zhong
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應用數學系
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Department of Applied Mathematics
Let omega(i) is an element of C (1 <= i <= n) and I is an element of S-n, the symmetric group of all permutations of 1, 2,...,n. Suppose A(I) is the weighted cyclic matrix [GRAPHICS] and omega(A(I)) denotes its numerical radius. We characterize those zeta is an element of S-n which satisfy omega(A(zeta)) = max(vertical bar is an element of Sn) omega(A(I)). The characterizations for unilateral and bilateral weighted (backward) shifts are also obtained. Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.
MINKOWSKI PRODUCT OF CONVEX SETS AND PRODUCT NUMERICAL RANGE
(2016-12)
Li, Chi-Kwong
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Pelejo, Diane Christine
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Poon, Yiu-Tung
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Wang, Kuo-Zhong
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應用數學系
Let K-1, K-2 be two compact convex sets in C. Their Minkowski product is the set K1K2 = {ab : a is an element of K-1, b is an element of K-2}. We show that the set K1K2 is star-haped if K-1 is a line segment or a circular disk. Examples for K-1 and K-2 are given so that K-1 and K-2 are triangles (including interior) and K1K2 is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science. Additional results and open problems are presented.
MINKOWSKI PRODUCT OF CONVEX SETS AND PRODUCT NUMERICAL RANGE
(2016-12)
Li, Chi-Kwong
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Pelejo, Diane Christine
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Poon, Yiu-Tung
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Wang, Kuo-Zhong
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應用數學系
Let K-1, K-2 be two compact convex sets in C. Their Minkowski product is the set K1K2 = {ab : a is an element of K-1, b is an element of K-2}. We show that the set K1K2 is star-haped if K-1 is a line segment or a circular disk. Examples for K-1 and K-2 are given so that K-1 and K-2 are triangles (including interior) and K1K2 is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science. Additional results and open problems are presented.
Numerical radii for tensor products of matrices
(2015-10-03)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
For n-by-n and m-by-m complex matrices A and B, it is known that the inequality w(A circle times B) = parallel to A parallel to w(B) holds, where w(center dot) and parallel to center dot parallel to denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if parallel to A parallel to = 1 and w(A circle times B) = w(B), then one of the following two conditions holds: (i) A has a unitary part, and (ii) A is completely nonunitary and the numerical range W(B) of B is a circular disc centered at the origin, (2) if parallel to A parallel to = parallel to A(k)parallel to = 1 for some k, 1 <= k < infinity, then w(A) >= cos(pi/(k + 2)), and, moreover, the equality holds if and only if A is unitarily similar to the direct sum of the (k + 1)-by-(k + 1) Jordan block J(k+1) and a matrix B with w(B) <= cos(pi/(k + 2)), and (3) if B is a nonnegative matrix with its real part (permutationally) irreducible, then w(A circle times B) = parallel to A parallel to w(B), if and only if either p (A) = infinity or n (B) = p (A) < infinity and B is permutationally similar to a block-shift matrix [GRAPHICS] with k = n (B), where p (A) = sup{l >= 1 : parallel to A(l)parallel to = parallel to A parallel to(l)} and n (B) = sup{l >= 1 : B-l not equal 0}.
Numerical Radii for Tensor Products of Operators
(2014-03-01)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
For bounded linear operators A and B on Hilbert spaces H and K, respectively, it is known that the numerical radii of A, B and are related by the inequalities . In this paper, we show that (1) if , then w(A) = rho(A) or w(B) = rho(B), where rho(center dot) denotes the spectral radius of an operator, and (2) if A is hyponormal, then . Here (2) confirms a conjecture of Shiu's and is proven via dilating the hyponormal A to a normal operator N with the spectrum of N contained in that of A. The latter is obtained from the Sz.-Nagy-FoiaAY dilation theory.
Numerical ranges and Gersgorin discs
(2013-02-01)
Chang, Chi-Tung
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Gau, Hwa-Long
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Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
For a complex matrix A = vertical bar a(ij)vertical bar(n)(i,j=1), let W(A) be its numerical range, and let G(A) be the convex hull of U-i=1(n) {z is an element of C : vertical bar z - a(ij)vertical bar <= (Sigma(i not equal j)(vertical bar a(ij)vertical bar)/2} and G'(A) = n{G(U*AU) : U n-by-n unitary). It is known that W(A) is always contained in G(A) and hence in G'(A). In this paper, we consider conditions for W (A) to be equal to G(A) or G'(A). We show that if W(A) = G'(A). then the boundary of W(A) consists only of circular arcs and line segments. If, moreover, A is unitarily irreducible, then W (A) is a circular disc. (Almost) complete characterizations of 2-by-2 and 3-by-3 matrices A for which W(A) = G'(A) are obtained. We also give criteria for the equality of W(A) and G(A). In particular, such A's among the permutationally irreducible ones must have even sizes. We also characterize those A's with size 2 or 4 which satisfy W(A) = G(A). Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.
Numerical ranges of row stochastic matrices
(2016-10-01)
Gau, Hwa-Long
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Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
In this paper, we consider properties of the numerical range of an n-by-n row stochastic matrix A. It is shown that the numerical radius of A satisfies 1 <= w(A) <= (1 + root n)/2, and, moreover, w(A) = 1 (resp., w(A) = (1 + root n)/2) if and only if A is doubly stochastic (resp., A = [GRAPHICS] for some j, 1 <= j <= n). A complete characterization of the A\'s for which the zero matrix of size n - 1 can be dilated to A is also given. Finally, for each n >= 2, we determine the smallest rectangular region in the complex plane whose sides are parallel to the x- and y-axis and which contains the numerical ranges of all n-by-n row stochastic matrices. (C) 2016 Elsevier Inc. All rights reserved.
NUMERICAL RANGES OF THE PRODUCT OF OPERATORS
(2017-03-01)
Du, Hongke
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Li, Chi-Kwong
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Wang, Kuo-Zhong
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Wang, Yueqing
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Zuo, Ning
We study containment regions of the numerical range of the product of operators A and B such that W(A) and W(B) are line segments. It is shown that the containment region is equal to the convex hull of elliptical disks determined by the spectrum of AB, and conditions on A and B for the set equality holding are obtained. The results cover the case when A and B are self-adjoint operators extending the previous results on the numerical range of the product of two orthogonal projections.
Numerical ranges of weighted shifts
(2011-09-15)
Wang, Kuo-Zhong
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
Let A be a unilateral (resp., bilateral) weighted shift with weights w(n), n >= 0 (resp., -infinity < n < infinity). Eckstein and Racz showed before that A has its numerical range W (A) contained in the closed unit disc if and only if there is a sequence {a(n)}(n=0)(infinity) (resp., {a(n))(n=-infinity)(infinity)) in [-1,1] such that |w(n)|(2) = (1 - a(n))(1 + a(n+1)) for all n. In terms of such a(n)'s, we obtain a necessary and sufficient condition for W (A) to be open. If the w(n)'s are periodic, we show that the a(n)'s can also be chosen to be periodic. As a result, we give an alternative proof for the openness of W (A) for an A with periodic weights, which was first proven by Stout. More generally, a conjecture of his on the openness of W (A) for A with split periodic weights is also confirmed. (C) 2011 Elsevier Inc. All rights reserved.
Operator norms and lower bounds of generalized Hausdorff matrices
(2011)
Chen, Chang-Pao
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Wang, Kuo-Zhong
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應用數學系
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Department of Applied Mathematics
Let A (a(n,k))(n,k >= 0) be a non-negative matrix. Denote by L(p,q)(A) the supremum of those L satisfying the following inequality: [GRAPHICS] The purpose of this article is to establish a Bennett-type formula for parallel to H(mu)(0)parallel to(down arrow)(p,p) and a Hardy-type formula for L(p,p)(down arrow)(H(mu)(alpha)) and L(p,p)H((alpha)(mu)), where H(mu)(alpha) is a generalized Hausdorff matrix and 0 < p <= 1. Similar results are also established for L(p,p)(H(mu)(alpha)) and L(p,p)H(((alpha)(mu))(t)) for other ranges of p and q. Our results extend [ Chen and Wang, Lower bounds of Copson type for Hausdorff matrices, Linear Algebra Appl. 422 ( 2007), pp. 208-217] and [ Chen and Wang, Lower bounds of Copson type for Hausdorff matrices: II, Linear Algebra Appl. 422 ( 2007) pp. 563-573] from H(mu)(0) to H(mu)(alpha) with alpha >= 0 and completely solve the value problem of parallel to H(mu)(0)parallel to(down arrow)(p,p), L(p,p)(down arrow)(H(mu)(alpha)), L(p,p)H((alpha)(mu)) and L(p,p)H(((alpha)(mu))(t)) for alpha is an element of N boolean OR {0}.