UB-matrices and conditions for Poncelet polygon to be closed

Mirman, Boris

doi:10.1016/s0024-3795(02)00447-0

Cited by 30 publications

(28 citation statements)

References 6 publications

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“…then K is finite-dimensional and S can be represented as an n × n matrix. Because much is known about the numerical ranges of these S [12,20,21,28], it is natural to consider Crouzeix's conjecture for them. Moreover, Sz.-Nagy and Foias [35] showed that every completely non-unitary contraction of class C 0 with defect 1 is unitarily equivalent to some S .…”

Section: Motivationmentioning

confidence: 99%

Crouzeix’s Conjecture and Related Problems

Bickel

Gorkin

Greenbaum

et al. 2020

Comput. Methods Funct. Theory

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Crouzeix's conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f (A) satisfies the estimate f (A) ≤ 2 sup{| f (z)| : z ∈ W (A)}, (1) where W (A) := { Ax, x : x = 1} denotes the numerical range of A. This would then also hold for all functions f which are analytic in a neighborhood of W (A). We provide a survey of recent investigations related to this conjecture and derive bounds for f (A) for specific classes of operators A. This allows us to state explicit conditions that guarantee that Crouzeix's estimate (1) holds. We describe properties of related extremal functions (Blaschke products) and associated extremal vectors. The case where A is a matrix representation of a compressed shift operator is studied in some detail.

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Section: Motivationmentioning

confidence: 99%

Crouzeix’s Conjecture and Related Problems

Bickel

Gorkin

Greenbaum

et al. 2020

Comput. Methods Funct. Theory

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“…In general, the defining equation of this envelope is not always reducible, but the "outermost part" of the curve gives the so-called Poncelet curve associated with the Blaschke product. For instance, see [Mir03]…”

Section: Interior and Exterior Curvesmentioning

confidence: 99%

Interior and exterior curves of finite Blaschke products

Fujimura

2018

Journal of Mathematical Analysis and Applications

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For a Blaschke product B of degree d and λ on ∂D, let λ be the set of lines joining each distinct two preimages in B −1 (λ). The envelope of the family of lines { λ } λ∈∂D is called the interior curve associated with B. In 2002, Daepp, Gorkin, and Mortini proved the interior curve associated with a Blaschke product of degree 3 forms an ellipse. While let L λ be the set of lines tangent to ∂D at the d preimages B −1 (λ) and the trace of the intersection points of each two elements in L λ as λ ranges over the unit circle is called the exterior curve associated with B. In 2017, the author proved the exterior curve associated with a Blaschke product of degree 3 forms a non-degenerate conic.In this paper, for a Blaschke product of degree d, we give some geometrical properties that lie between the interior curve and the exterior curve.

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“…We now compare Theorem 2 to the following. The connection between these two theorems has been well studied; in particular, Gau and Wu's work [7][8][9][10]26] and Mirman's papers [18][19][20][21][22] tie these theorems together using Poncelet's theorem.…”

Section: Theorem 2 (Seementioning

confidence: 99%