Constraint Unitary Dilations and Numerical Ranges
نویسندگان
چکیده
It is shown that each contraction A on a Hilbert space H, with A + A I for some 2 R, has a unitary dilation U on H H satisfying U + U I. This is used to settle a conjecture of Halmos in the aarmative: The closure of the numerical range of each contraction A is the intersection of the closures of the numerical ranges of all unitary dilations of A. By means of the duality theory of completely positive linear maps, some further results concerning numerical ranges inclusions and dilations are deduced.
منابع مشابه
Higher-rank Numerical Ranges and Dilations
For any n-by-n complex matrix A and any k, 1 ≤ k ≤ n, let Λk(A) = {λ ∈ C : X∗AX = λIk for some n-by-k X satisfying X∗X = Ik} be its rank-k numerical range. It is shown that if A is an n-by-n contraction, then Λk(A) = ∩{Λk(U) : U is an (n + dA)-by-(n + dA) unitary dilation of A}, where dA = rank (In − A∗A). This extends and refines previous results of Choi and Li on constrained unitary dilations...
متن کاملProperties of matrices with numerical ranges in a sector
Let $(A)$ be a complex $(ntimes n)$ matrix and assume that the numerical range of $(A)$ lies in the set of a sector of half angle $(alpha)$ denoted by $(S_{alpha})$. We prove the numerical ranges of the conjugate, inverse and Schur complement of any order of $(A)$ are in the same $(S_{alpha})$.The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product, star-congruen...
متن کاملDilations for $C^ast$-dynamical systems with abelian groups on Hilbert $C^ast$-modules
In this paper we investigate the dilations of completely positive definite representations of (C^ast)-dynamical systems with abelian groups on Hilbert (C^ast)-modules. We show that if ((mathcal{A}, G,alpha)) is a (C^ast)-dynamical system with (G) an abelian group, then every completely positive definite covariant representation ((pi,varphi,E)) of ((mathcal{A}, G,alpha)) on a Hilbert ...
متن کاملHigher–rank Numerical Ranges of Unitary and Normal Matrices
We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary matrix are given by complex polygons determined by the spectral structure of the matrix. We discuss applications of the results to quantum error correction, s...
متن کاملMatrices with Circular Symmetry on Their Unitary Orbits and C-numerical Ranges
We give equivalent characterizations for those n x n complex matrices A whose unitary orbits %?(A) and C-numerical ranges WC{A) satisfy ei8&(A) = f/(A) or e'e WC(A) = WC(A) for some real 0 (or for all real 0 ). In particular, we show that they are the block-cyclic or block-shift operators. Some of these results are extended to infinite-dimensional Hubert spaces.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1999