Browsing by Author "Wu, PY"

Now showing 1 - 18 of 18

Band-diagonal operators
(1996-11-15)
Fong, CK
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Wu, PY
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應用數學系
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Department of Applied Mathematics
We show that there are operators on a five-dimensional Hilbert space which are not tridiagonal, and that there are compact operators and normal operators on separable infinite-dimensional spaces which are not band-diagonal.
Condition for the numerical range to contain an elliptic disc
(2003-05-01)
Gau, HL
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Wu, PY
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應用數學系
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Department of Applied Mathematics
For an n-by-n matrix A and an elliptic disc E in the plane, we show that the sum of the number of common supporting lines and the number of common intersection points to E and the numerical range W (A) of A should be at least 2n + 1 in order to guarantee that E be contained in W(A). This generalizes previous results of Anderson and Thompson. As an application, our result is used to verify a special case of the Poncelet property conjecture. (C) 2003 Elsevier Science Inc. All rights reserved.
Dilation to the unilateral shifts
(1998-09-01)
Takahashi, K
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Wu, PY
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應用數學系
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Department of Applied Mathematics
The classical result of Foias says that an operator power dilates to a unilateral shift if and only if it is a C-.o contraction. In this paper, we consider the corresponding question of dilating to a unilateral shift. We show tht for contractions with, at least one defect index finite, dilation and power dilation to some unilateral shift amount to the same thing. The only difference is on the minimum multiplicity of the unilateral shift to which the contraction can be (power) dilated. We also obtain a characterization of contractions which are finite-rank perturbations of a unilateral shift, generalizing the rank-one perturbation result of Nakamura.
Dilation to unilateral shifts
(2000)
Wu, PY
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Takahashi, K
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應用數學系
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Department of Applied Mathematics
Direct sums of irreducible operators
(2003)
Fang, JS
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Jiang, CL
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Wu, PY
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應用數學系
;
Department of Applied Mathematics
It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators T which are direct sums of irreducible operators in terms of the C*-structure of the commutant of the von Neumann algebra generated by T.
Finite Blaschke products of contractions
(2003-07-15)
Gau, HL
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Wu, PY
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應用數學系
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Department of Applied Mathematics
Let A be a contraction on Hilbert space H and phi a finite Blaschke product. In this paper, we consider the problem when the norm of phi(A) is equal to 1. We show that (1) parallel tophi(A)parallel to = 1 if and only if parallel toA(k)parallel to = 1, where k is the number of zeros of phi counting multiplicity, and (2) if H is finite-dimensional and A has no eigenvalue of modulus 1, then the largest integer I for which parallel toA(l)parallel to = 1 is at least m/(n - m), where n = dim H and m = dim ker(I - A*A), and, moreover, l = n - 1 if and only if m = n - 1. (C) 2003 Elsevier Science Inc. All rights reserved.
Finite-rank perturbations of positive operators and isometries
(2006)
Choi, MD
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Wu, PY
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應用數學系
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Department of Applied Mathematics
We completely characterize the ranks of A - B and A(1/2) - B-1/2 for operators A and B on a Hilbert space satisfying A >= B >= 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and m = rank(A(1/2) - B-1/2) for some operators A and B with A >= B >= 0 on a Hilbert space of dimension n (1 <= n <= infinity) if and only if l = m = 0 or 0 < l <= m <= n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that of rank(A(1/2) - B-1/2). For two isometries, we give necessary and sufficient conditions in order that they be finite-rank perturbations of each other. One such condition says that, for isometries A and B, A - B has finite rank if and only if A = (I + F)B for some unitary operator I + F with finite-rank F. Another condition is in terms of the parts in the Wold-Lebesgue decompositions of the nonunitary isometries A and B.
Higher-dimensional numerical ranges of quadratic operators
(2003-12-01)
Chien, MT
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Tso, SH
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Wu, PY
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應用數學系
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Department of Applied Mathematics
We show that for every positive integer k, the k-numerical range of a square-zero operator on a (separable) Hilbert space is an (open or closed) circular disc centered at the origin. The radius and the closedness of the disc can be completely determined in terms of the "singular numbers" of the operator. The k-numerical range of idempotent operators is more difficult to describe since its boundary is in general not any familiar curve. What we do is to give enough information, again in terms of the singular numbers of the idempotent operator under consideration, so as to have a general idea of its shape and location.
Matricial ranges of quadratic operators
(1999-09-01)
Tso, SH
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Wu, PY
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應用數學系
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Department of Applied Mathematics
We show that if T is a quadratic operator on a Hilbert space, then (1) the numerical range of T is an (open or closed) elliptical disc (or its degenerate form) and (2) for every n greater than or equal to 1, the nth matricial range of T consists of n x n matrices whose numerical ranges are contained in the closure of the numerical range of T.
Numerical range and Poncelet property
(2003-06-01)
Gau, HL
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Wu, PY
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應用數學系
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Department of Applied Mathematics
In this survey article, we give an expository account of the recent developments on the Poncelet property for numerical ranges of the compressions of the shift S(phi). It can be considered as an updated and more advanced edition of the recent expository article published in the American Mathematical Monthly by the second author on this topic. The new information includes: (1) a simplified approach to the main results (generalizations of Poncelet, Brianchon-Ceva and Lucas-Siebeck theorems) in this area, (2) the recent discovery of Mirman reftiting a previous conjecture on the coincidence of Poncelet curves and boundaries of the numerical ranges of finite-dimensional S(phi), and (3) some partial generalizations by the present authors of the above-mentioned results from the unitary-dilation context to the normal-dilation one and also from the finite-dimensional S(phi) to the infinite-dimensional.
Numerical range circumscribed by two polygons
(2004-05-01)
Gau, HL
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Wu, PY
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應用數學系
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Department of Applied Mathematics
We show that, for any 2n+2 distinct points a(1), a'(1), a(2), a'(2),...,a(n+1), a'(n+1) (in this order) on the unit circle, there is an n-by-n matrix A. unique LIP to Unitary equivalence, which has norm one and satisfies the conditions that it has all its eigenvalues in the open unit disc, I-n-A*A has rank one and its numerical range is circumscribed by the two (n+1)-gons a(1)a(2)...a(n+1) and a'(1)a'(2)...a'(n+1). This generalizes the classical result of the existence of a conical Curve Circumscribed by two triangles which are already inscribed on another conical curve. (C) 2004 Elsevier Inc. All rights reserved.
Numerical range of a normal compression
(2004-05-01)
Gau, HL
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Wu, PY
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應用數學系
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Department of Applied Mathematics
Let N be an n-by-n diagonal matrix whose distinct eigenvalues form corners of their convex hull, let x be a vector in with nonzero components, and let A be the compression of N to the orthogonal complement of x . In this article, we study the properties of the eigenvalues and the numerical range of A and show that in many ways they are analogous to the ones for unitary N . The approach via the diagonal form of N yields a much simpler proof for some of the main results in this area.
Numerical range of a normal compression II
(2004-10-01)
Gau, HL
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Wu, PY
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應用數學系
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Department of Applied Mathematics
As in the predecessor [Numerical range of a normal compression, Linear and Multilinear Algebra, in press] of this paper, we consider properties of matrices of the form V*NV, where N = diag(a(1),..., a(n+1)) is a diagonal matrix with distinct eigenvalues a(j)s such that each of them is a corner of the convex hull they generate, and V is an (n + 1)-by-n matrix with V*V = I-n such that any nonzero vector orthogonal to the range space of V has all its components nonzero. We obtain that such a matrix A is determined by its eigenvalues up to unitary equivalence, is irreducible and cyclic, and the boundary of its numerical range is a differentiable curve which contains no line segment. We also consider the condition for the existence of another matrix of the above type which dilates to A such that their numerical ranges share some common points with the boundary of the (n + 1)-gon a(1) (...) a(n+1). (C) 2004 Elsevier Inc. All rights reserved.
Numerical range of Aluthge transform of operator
(2002-12-15)
Wu, PY
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應用數學系
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Department of Applied Mathematics
For any operator A on a Hilbert space, let (A) over tilde denote its Aluthge transform. In this paper, we prove that the closure of the numerical range of (A) over tilde is always contained in that of A. This supplements the recently proved case for dim ker A less than or equal to dim ker A* by Yamazaki, and partially confirms a conjecture of Jung, Ko and Pearcy. (C) 2002 Elsevier Science Inc. All rights reserved.
Singular unitary dilations
(1999-02-01)
Wu, PY
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Takahashi, K
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應用數學系
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Department of Applied Mathematics
We study the problem of determining which bounded linear operator on a Hilbert space can be dilated to a singular unitary operator. Same of the partial results we obtained are (1) every strict contraction has a diagonal unitary dilation, (2) every C-0 contraction has a singular unitary dilation, and (3) a contraction with one of its defect indices finite has a singular unitary dilation if and only if it is the direct sum of a singular unitary operator and a C-0(N) contraction. Such results display a scenario which is in marked contrast to that of the classical case where we have the absolute continuity of the minimal unitary power dilation of any completely nonunitary contraction.
Sums of strongly irreducible operators
(1998-01-01)
Jiang, CL
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Wu, PY
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應用數學系
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Department of Applied Mathematics
Sums of strongly irreducible operators
(1998-01-01)
Jiang, CL
;
Wu, PY
;
應用數學系
;
Department of Applied Mathematics
The numerical range of a nonnegative matrix
(2002-07-15)
Li, CW
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Tam, BS
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Wu, PY
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應用數學系
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Department of Applied Mathematics
We offers an almost self-contained development of Perron-Frobenius type results for the numerical range of an (irreducible) nonnegative matrix, rederiving and completing the previous work of Issos, Nylen and Tam, and Tam. and Yang on this topic. We solve the open problem of characterizing nonnegative matrices whose numerical ranges are re gular convex polygons with center at the origin. Some related results are obtained and some open problems are also posed. (C) 2002 Elsevier Science Inc. All rights reserved.