Browsing by Author "Wu, Pei Yuan"

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Anderson's theorem for compact operators
(2006)
Gau, Hwa-Long
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Wu, Pei Yuan
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應用化學系
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Department of Applied Chemistry
It is shown that if A is a compact operator on a Hilbert space with its numerical range W(A) contained in the closed unit disc (D) over bar and with W(A) intersecting the unit circle at infinitely many points, then (WA) over bar is equal to (D) over bar. This is an infinite-dimensional analogue of a result of Anderson for finite matrices.
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
;
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.
Defect indices of powers of a contraction
(2010-06-01)
Gau, Hwa-Long
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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 contraction on a Hilbert space H. The defect index d(A) of A is, by definition, the dimension of the closure of the range of l - A*A. We prove that (1) d(An) <= nd(A) for all n >= 0, (2) if, in addition, A(n) converges to 0 in the strong operator topology and d(A) = 1, then d(An) = n for all finite n, 0 <= n <= dim H, and (3) d(A) = d(A)* implies d(An) = d(An)* for all n >= 0. The norm-one index k(A) of A is defined as sup{n >= 0 : parallel to A(n)parallel to = 1}. When dim H = m < infinity, a lower bound for k(A) was obtained before: k(A) >= (m/d(A)) - 1. We show that the equality holds if and only if either A is unitary or the eigenvalues of A are all in the open unit disc, d(A) divides m and d(An) = nd(A) for all n, 1 <= n <= m/d(A). We also consider the defect index of f(A) for a finite Blaschke product f and show that d(f(A)) = d(An), where n is the number of zeros off. (C) 2009 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 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
;
應用數學系
;
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.
Equality of three numerical radius inequalities
(2018-10-01)
Gau, Hwa-Long
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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, let w(A) and parallel to A parallel to denote its numerical radius and operator norm, respectively. The following three inequalities, each a strengthening of w(A) <= parallel to A parallel to, are known to hold: w(A)(2) <= (parallel to A parallel to(2) + w(A(2)))/2, w(A) <= (parallel to A parallel to + parallel to A(2)parallel to(1/2))/2, and w(A) <= (parallel to A parallel to + w(Delta(t)(A)))/2 (0 <= t <= 1), where Delta(t)(A) is the generalized Aluthge transform of A. In this paper, we derive necessary and sufficient conditions in terms of the operator structure of A for which the inequalities become equalities. (C) 2018 Elsevier Inc. All rights reserved.
Extremality of numerical radii of matrix products
(2016-07-15)
Gau, Hwa-Long
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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 two n-by-n matrices A and B, it was known before that their numerical radii satisfy the inequality w(AB) <= 4w(A)w(B), and the equality is attained by the 2-by-2 matrices A = [GRAPHICS] and B = [GRAPHICS] . Moreover, the constant "4" here can be reduced to "2" if A and B commute, and the corresponding equality is attained by A = I-2 circle times [GRAPHICS] and B = [GRAPHICS] circle times I-2. In this paper, we give a complete characterization of A and B for which the equality holds in each case. More precisely, it is shown that w(AB) = 4w(A)w(B) (resp., w(AB) = 2w(A)w(B) for commuting A and B) if and only if either A or B is the zero matrix, or A and B are simultaneously unitarily similar to matrices of the form [GRAPHICS] circle plus A\' and [GRAPHICS] circle plus B\' (resp., [GRAPHICS] circle plus A\' and [GRAPHICS] circle plus B\') with w(A\') <= vertical bar a vertical bar/2 and w(B\') <= vertical bar b vertical bar/2. An analogous characterization for the extremal equality for tensor products is also proven. For doubly commuting matrices, we use their unitary similarity model to obtain the corresponding result. For commuting 2-by-2 matrices A and B, we show that w(AB) = w(A)w(B) if and only if either A or B is a scalar matrix, or A and B are simultaneously unitarily similar to [GRAPHICS] and [GRAPHICS] with vertical bar a(1)vertical bar >= vertical bar a(2)vertical bar and vertical bar b(1)vertical bar >= vertical bar b(2)vertical bar. (C) 2016 Elsevier Inc. All rights reserved.
Fredholmness of linear combinations of two idempotents
(2007-12-01)
Gau, Hwa-Long
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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 E and F be idempotent operators on a complex Hilbert space, and lot a and b be nonzero scalars with a + b not equal 0. We prove that aE + bF is Fredholm if and only if E + F is, thus answering affirmatively a question asked by Koliha and Rakocevic.
HIGHER-RANK NUMERICAL RANGES AND DILATIONS
(2010-12-01)
Gau, Hwa-Long
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Li, Chi-Kwong
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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 complex matrix A and any k, 1 <= k <= n, let Lambda(k)(A) = {lambda is an element of C : X*AX = lambda I(k) for some n-by-k X satisfying X*X = I(k)) be its rank-k numerical range. It is shown that if A is an n-by-n contraction, then Lambda(k)(A) = boolean AND{Lambda(k)(U) : U is an (n + d(A))-by-(n + d(A)) unitary dilation of A}, where d(A) = rank (I(n) - A* A). This extends and refines previous results of Choi and Li on constrained unitary dilations, and a result of Mirman on S(n)-matrices.
Higher-rank numerical ranges and Kippenhahn polynomials
(2013-04-01)
Gau, Hwa-Long
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
We prove that two n-by-n matrices A and B have their rank-k numerical ranges Lambda(k) (A) and Lambda(k) (B) equal to each other for all k, 1 <= k <= left perpendicularn/2right perpendicular + 1, if and only if their Kippenhahn polynomials P-A (x, y, z) equivalent to det(xRe A + yIm A + zI(n)) and p(B) (x, y, z) equivalent to det(xRe B + yIm B + zI(n)) coincide. The main tools for the proof are the Li-Sze characterization of higher-rank numerical ranges, Weyl's perturbation theorem for eigenvalues of Hermitian matrices and Bezout's theorem for the number of common zeros for two homogeneous polynomials. (C) 2012 Elsevier Inc. All rights reserved.
Inner functions of numerical contractions
(2009-04-15)
Gau, Hwa-Long
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
We prove that, for a function f in H(infinity) of the unit disc with parallel to f parallel to(infinity) <= 1, the existence of an operator T on a complex Hilbert space H with its numerical radius at most one and with parallel to f(T)x parallel to = 2 for some unit vector x in H is equivalent to that f be an inner function with f (0) = 0. This confirms a conjecture of Drury [S.W. Drury, Symbolic calculus of operators with unit numerical radius, Linear Algebra Appl. 428 (2008) 2061-2069]. Moreover, we also show that any operator T satisfying the above conditions has a direct summand similar to the compression of the shift S(phi), where phi(z) = zf(z) for vertical bar z vertical bar < 1. This generalizes the result of Williams and Crimmins [J.P. Williams, T. Crimmins, On the numerical radius of a linear operator, Amer. Math. Monthly 74 (1967) 832-833] for f (z) = z and of Crabb [M.J. Crabb, The powers of an operator of numerical radius one, Michigan Math. J. 18 (1971) 253-256] for f(z) = z(n) (n >= 2). (C) 2008 Elsevier Inc. All rights reserved.
Line segments and elliptic arcs on the boundary of a numerical range
(2008)
Gau, Hwa-Long
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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 the numbers of line segments and elliptic arcs on the boundary partial derivative W(A) of its numerical range. We show that (a) if n >= 4 and A has an (n-1)-by-(n-1) submatrix B with W(B) an elliptic disc, then there can be at most 2n-2 line segments on partial derivative W(A), and (b) if n >= 3, then partial derivative W(A) contains at most (n-2) arcs of any ellipse. Moreover, both upper bounds are sharp. For nilpotent matrices, we also obtain analogous results with sharper bounds.
LOWER BOUNDS FOR THE NUMERICAL RADIUS
(2017-12-01)
Gau, Hwa-Long
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
C We show that if A = [a(ij)](i)(n),(j=1) is an n-by-n complex matrix and A '= [a '(ij)](i)(n),(j=1), where a '(ij) ={a(ij) if (i,j) = (1,2),..., (n-1,n) or (n,1), 0 otherwise, then w(A) >= w(A '), where w((.)) denotes the numerical radius of a matrix. Moreover, if n is odd and a(12),..., a(n-1),(n),a(n1) are all nonzero, then w(A) = w(A ') if and only if A = A '. For an even n, under the same nonzero assumption, we have W(A) = W(A ') if and only if A = A ', where W((.)) is the numerical range of a matrix.
Noncircular elliptic discs as numerical ranges of nilpotent operators
(2012)
Gau, Hwa-Long
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Wu, Pei Yuan
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應用數學系
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Department of Applied Mathematics
We show that (1) if A is a nonzero quasinilpotent operator with ran A(n) closed for some n >= 1, then its numerical range W(A) contains 0 in its interior and has a differentiable boundary, and (2) a noncircular elliptic disc can be the numerical range of a nilpotent operator with nilpotency 3 on an infinite-dimensional separable space. (1) is a generalization of the known result for nonzero nilpotent operators, and (2) is in contrast to the finite-dimensional case, where the only elliptic discs which are the numerical ranges of nilpotent finite matrices are the circular ones centred at the origin.
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.