Browsing by Author "Takahashi, K"

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Dilation to the unilateral shifts
(1998-09-01)
Takahashi, K
;
Wu, PY
;
應用數學系
;
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
;
Takahashi, K
;
應用數學系
;
Department of Applied Mathematics
On sums of three square-zero matrices
(2000-02-15)
Takahashi, K
;
應用數學系
;
Department of Applied Mathematics
Wang and Wu characterized matrices which are sums of two square-zero matrices, and proved that every matrix with trace-zero is a sum of four square-zero matrices. Moreover, they gave necessary or sufficient conditions for a matrix to be a sum of three square-zero matrices. In particular, they proved that if an n x n matrix A is a sum of three square-zero matrices, the dim ker(A - alpha I) less than or equal to 3n/4 for any scalar alpha not equal 0. Proposition 1 shows that this condition is not necessarily sufficient for the matrix A to be a sum of three square-zero matrices, and characterizes sums of three square-zero matrices among matrices with minimal polynomials of degree 2. (C) 2000 Elsevier Science Inc, All rights reserved.
Singular unitary dilations
(1999-02-01)
Wu, PY
;
Takahashi, K
;
應用數學系
;
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.