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Numerical range of s(φ)
We make a detailed study of the numerical ranges W(T) of completely nonunitary contractions T with the property rank (1-T∗T)=1 on a finite-dimensional Hilbert space. We show that such operators areExpand
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 anExpand
Condition for the numerical range to contain an elliptic disc
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(Expand
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-nExpand
Anderson’s theorem for compact operators
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 and with W(A) intersecting the unit circle at infinitely many points,Expand
Numerical Ranges of KMS Matrices
A KMS matrix is one of the form $$J_n(a)=[{array}{ccccc} 0 & a & a^2 &... & a^{n-1} & 0 & a & \ddots & \vdots & & \ddots & \ddots & a^2 & & & \ddots & a 0 & & & & 0{array}]$$ for $n\ge 1$ and $a$ inExpand
Companion matrices: reducibility, numerical ranges and similarity to contractions
Abstract In this paper, we study some unitary-equivalence properties of the companion matrices. We obtain a criterion for a companion matrix to be reducible and show that the numerical range of aExpand
Lucas' theorem refined ∗
We prove a refined version of the classical Lucas' theorem: if p is a polynomial with zeros a 1,…,a n+1 all having modulus one and φis the Blaschke product whose zeros are those of the derivative pExpand
Numerical range circumscribed by two polygons
Abstract We show that, for any 2 n +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 up to unitaryExpand
Numerical Range of a Normal Compression
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 orthogonalExpand