Constants related to operators of class C?

Okubo, Kimihiro; Andô, Tsuyoshi

doi:10.1007/bf01323467

Cited by 34 publications

(27 citation statements)

References 2 publications

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“…As we mentioned in the introduction, Crouzeix showed [6] that the numerical range W (T ) is a complete C-spectral set for T for a universal constant C < 11.08. Crouzeix conjectures [5] that the optimal constant is 2, which is the case for a disc by [10]. The following are immediate from Theorem 3.1.…”

Section: Consequencesmentioning

confidence: 78%

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Complete spectral sets and numerical range

Davidson

Paulsen

Woerdeman

2017

Proc. Amer. Math. Soc.

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Abstract. We define the complete numerical radius norm for homomorphisms from any operator algebra into B(H), and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if K is a complete Cspectral set for an operator T , then it is a complete M -numerical radius set, where M = 1 2 (C + C −1 ). In particular, in view of Crouzeix's theorem, there is a universal constant M (less than 5.6) so that if P is a matrix polynomial and T ∈ B(H), thenIn 2007, Michel Crouzeix [6] proved the remarkable fact that for any operator T on a Hilbert space H, the numerical range is a complete C-spectral set for some constant with a universal bound of 11.08. Moreover in [5], he conjectures that the optimal constant is 2, which is the case for a disc. This inspired a result of Drury [7] who proved that if the numerical range of T is contained in the disc, then the numerical radius of any polynomial in T is bounded by 5 4 times the supremum norm of the polynomial over the disc. Generally all that one can say about the relationship between the norm and numerical radius is that w(X) ≤ X , with equality for many operators, so the improvement from 2 to Drury's 5 4 was unexpected. In this note, we establish a precise relationship between the completely bounded norm of a homomorphism of an arbitrary operator algebra and what we call the complete numerical radius norm of the homomorphism. When applied to the case of the disc, our relationship yields Drury's result in the matrix polynomial case, and our result also applies to the general study of C-spectral sets.2010 Mathematics Subject Classification. Primary 47A12, 47A25, 15A60.

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Section: Consequencesmentioning

confidence: 78%

“…By [10,Theorem 2], there is an invertible operator S such that S −1 T S ≤ 1 and S S −1 ≤ ρ. After scaling, we may suppose that F ∞ = 1.…”

Section: Consequencesmentioning

confidence: 99%

Complete spectral sets and numerical range

Davidson

Paulsen

Woerdeman

2017

Proc. Amer. Math. Soc.

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“…Crouzeix showed that in fact such a constant does exist and is between 2 and 11.08. Okubo and Ando [57] showed that the conjecture is true if the numerical range is a disk. Badea, Crouzeix and Delyon presented several approaches to this problem and others in [6].…”

Section: Open Questionsmentioning

confidence: 96%

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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“…Based on the results in [12], it has been shown in [13] that if the field of values W (A) is a disk, then (14) holds. Thus, for half-radial matrices we conclude the following.…”

Section: Consequences On Crouzeix's Conjecturementioning

confidence: 99%

Characterization of half-radial matrices

Hnětýnková

Tichý

2018

Linear Algebra and its Applications

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Numerical radius r(A) is the radius of the smallest ball with the center at zero containing the field of values of a given square matrix A. It is well known that r(A) ≤ A ≤ 2r(A), where · is the matrix 2-norm. Matrices attaining the lower bound are called radial, and have been analyzed thoroughly. This is not the case for matrices attaining the upper bound where only partial results are available. In this paper we consider matrices satisfying r(A) = A /2, and call them half-radial. We summarize the existing results and formulate new ones. In particular, we investigate their singular value decomposition and algebraic structure, and provide other necessary and sufficient conditions for a matrix to be half-radial. Based on that, we study the extreme case of the attainable constant 2 in Crouzeix's conjecture. The presented results support the conjecture of Greenbaum and Overton, that the Crabb-Choi-Crouzeix matrix always plays an important role in this extreme case.

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Constants related to operators of class C?

Cited by 34 publications

References 2 publications

Complete spectral sets and numerical range

Complete spectral sets and numerical range

Numerical range and compressions of the shift

Characterization of half-radial matrices

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