Spectral sets and operator radii

Badea, Cătălin; Crouzeix, Michel; Klaja, Hubert

doi:10.1112/blms.12191

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…As a w-extremal function is also of the formB A • φ for some Blaschke productB A with degree strictly less than n, it makes sense to ask: This is an interesting question, because there is a close connection between the extremal and w-extremal problems. It was shown in [3] that inequality (2) holds if and only if…”

Section: Proof Of Theorem 38mentioning

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: Proof Of Theorem 38mentioning

confidence: 99%

Crouzeix’s Conjecture and Related Problems

Bickel

Gorkin

Greenbaum

et al. 2020

Comput. Methods Funct. Theory

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show abstract

“…As a w-extremal function is also of the form BA • φ for some Blaschke product BA with degree strictly less than n, it makes sense to ask: This is an interesting question, because there is a close connection between the extremal and w-extremal problems. It was shown in [3] that inequality (1) holds if and only if…”

Section: Apparent Degree Of Extremal F For 3x3 Random Matricesmentioning

confidence: 99%

Crouzeix's Conjecture and related problems

Bickel,

Gorkin,

Greenbaum

et al. 2020

Preprint

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In this paper, we establish several results related to Crouzeixs conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.

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“…Remark 11. The equation ( 17) above was remarked without proof in [1] in the following, slightly different form, namely:…”

Section: Connection With the Classes C ρ Of Power Bounded Operatorsmentioning

confidence: 99%

Between the von Neumann inequality and the Crouzeix conjecture

Pagacz¹,

Pietrzycki²,

Wojtylak³

2019

Preprint

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A new concept of a deformed numerical range Wq(T ), where T is a bounded linear operator or a matrix and q ∈ [0, 2) is a parameter, is introduced. Each Wq(T ) is a closed convex set that contains the spectrum of T . Furthermore, Wq(T ) is decreasing with respect to q and W1(T ) is the numerical range. It is also shown that Wq(T ) is contained in the closed unit disc if and only if T has a 2/(2−q) unitary dilation in the sense of Nágy-Foiaş. Spectral constants of WqT are investigated.

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Spectral sets and operator radii

Cited by 5 publications

References 19 publications

Crouzeix’s Conjecture and Related Problems

Crouzeix’s Conjecture and Related Problems

Crouzeix's Conjecture and related problems

Between the von Neumann inequality and the Crouzeix conjecture

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