Abstract
Abstract In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂ n cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n. The proof is based on the unitary similarity of A to a compressed shift operator SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.
Highlights
In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on Cn cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤
We use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions
An operator A acting in Cn is a unitarily irreducible partial isometry with dim ker A = if and only if A is a non-invertible matrix of class Sn, which consists of the contractions A ∈ Cn×n with all eigenvalues in the unit disk D and rank(I − A*A) =
Summary
Denote by Cn×n the algebra of all n-by-n matrices with complex entries. The numerical range W(A) of A ∈ Cn×n is the set of values of the quadratic form Ax, x on the unit sphere of Cn. An operator A acting in Cn is a unitarily irreducible partial isometry with dim ker A = if and only if A is a non-invertible matrix of class Sn, which consists of the contractions A ∈ Cn×n with all eigenvalues in the unit disk D and rank(I − A*A) = (see [8, Proposition 2.3]). Gau and Wu prove that among all operators acting in Cn those in Sn are distinguished by the so-called Poncelet property of their numerical range: For each t on the unit circle T there exists a (n+ )-gon Pt which is inscribed in T, circumscribed about W(A), and has t as a vertex The vertices of these Poncelet polygons Pt are the eigenvalues of unitary dilations Ut of A ([9, Theorem 2.1], for an alternative proof see [5]). What remains to prove is that the Poncelet curve C(B ) can only be circular if it is centered at the origin
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