Numerical ranges of KMS matrices

Gau, Hwa Long; Wu, Pei Yuan

doi:10.1007/bf03651342

Cited by 4 publications

(4 citation statements)

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“…2 ) ⊆ W (S Θ ), we see that c ± iR must satisfy (27). So, with |y| = R in (27) we immediately obtain…”

Section: Pseudohyperbolic Disks and Numerical Rangesmentioning

confidence: 85%

“…The matrix A t is sometimes called a KMS matrix and the numerical ranges of these matrices have been studied by Gau and Wu in [27,28]. In the 3 × 3 case, Crouzeix's results from [14] can be applied to obtain the boundary of W (A t ) and hence, of W (M t ).…”

Section: The Case Of a Unicritical θmentioning

confidence: 99%

“…We show this leads to a contradiction for t > 3/4. First, note that z 0 ∈ W (S Θ ) and so (27) implies that…”

Section: Pseudohyperbolic Disks and Numerical Rangesmentioning

confidence: 99%

“…The statement of ( 2) suggests that we need to study the numerical range of the associated compression of the shift S Θ . For unicritical Θ, these numerical ranges were studied by Gaaya in [21,22], Gau and Wu in [27,28], and in work of Partington and the second author [30]. When deg Θ = 3, results about W (S Θ ) are encoded in Crouzeix's work [14]; indeed, his arguments imply that the full Crouzeix conjecture holds for S Θ when deg Θ = 3 and Θ is unicritical.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Blaschke Products, Level Sets, and Crouzeix's Conjecture

Bickel¹,

Gorkin²

2021

Preprint

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We study several problems motivated by Crouzeix's conjecture, which we consider in the special setting of model spaces and compressions of the shift with finite Blaschke products as symbols. We pose a version of the conjecture in this setting, called the level set Crouzeix (LSC) conjecture, and establish structural and uniqueness properties for (open) level sets of finite Blaschke products that allow us to prove the LSC conjecture in several cases. In particular, we use the geometry of the numerical range to prove the LSC conjecture for compressions of the shift corresponding to unicritical Blaschke products of degree 4.

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“…2 ) ⊆ W (S Θ ), we see that c ± iR must satisfy (27). So, with |y| = R in (27) we immediately obtain…”

Section: Pseudohyperbolic Disks and Numerical Rangesmentioning

confidence: 85%

Section: The Case Of a Unicritical θmentioning

confidence: 99%

“…We show this leads to a contradiction for t > 3/4. First, note that z 0 ∈ W (S Θ ) and so (27) implies that…”

Section: Pseudohyperbolic Disks and Numerical Rangesmentioning

confidence: 99%