Numerical ranges and Poncelet curves

Mirman, Boris

doi:10.1016/s0024-3795(98)10037-x

Cited by 48 publications

(25 citation statements)

References 5 publications

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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%

Poncelet's theorem, Sendov's conjecture, and Blaschke products

Daepp

Gorkin

Voss

2010

Journal of Mathematical Analysis and Applications

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Section: Theorem 2 (Seementioning

confidence: 99%

Poncelet's theorem, Sendov's conjecture, and Blaschke products

Daepp

Gorkin

Voss

2010

Journal of Mathematical Analysis and Applications

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“…see [31,32]. Their work-and that of Mirman in [55]-shows that each ∂W (S B ) also satisfies an elegant geometric condition called the Poncelet property.…”

Section: Introductionmentioning

confidence: 99%

Numerical range and compressions of the shift

Bickel¹,

Gorkin²

2020

Complex Analysis and Spectral Theory

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The numerical range of a bounded, linear operator on a Hilbert space is a set in C that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space H 2 (D), and compressions of the shift operator to model spaces, i.e. spaces of the form H 2 θH 2 where θ is inner. For these compressions of the shift operator, we provide a survey of results on the connection between their numerical ranges and the numerical ranges of their unitary dilations. We also discuss related results for compressed shift operators on the bidisk associated to rational inner functions and conclude the paper with a brief discussion of the Crouzeix conjecture. Primary 47A12; Secondary 47A13, 30C15 Date: October 30, 2018.

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“…The final theme concerns a class of finite dimensional matrices, now called S n , studied in a series of independent papers by Gau-Wu [24,25,26,27,28,29,30], also [76], and by Mirman [50,51,52,53,54] both series starting in 1998. Recall that an operator on a Hilbert space is called a contraction if its norm is at most 1.…”

Section: Introductionmentioning

confidence: 99%