On the normalized numerical range

Spitkovsky, Ilya M.; Stoica, Andrei-Florian

doi:10.7153/oam-11-15

Cited by 7 publications

(10 citation statements)

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“…In fact, the condition dim DW(A) ≤ 1 holds if and only if A is normal, with at most two distinct eigenvalues. In that case F N (A) is a hyperbolic arc, as for n = 2 was established in [10, Proposition 2.1], and observed to be valid for arbitrary n in [18,Theorem 5.6].…”

Section: Normal Casementioning

confidence: 84%

“…It is worth mentioning, however, that for non-invertible A the set F N (A) may not be closed. The respective examples exist even with A ∈ M 2 (C) and can be found in [9]; the closedness criterion is given by [18,Theorem 6.4]. Property (g) was proved in [18,Section 3], while the path-connectedness of F N (A) was established earlier in [6,Proposition 7].…”

Section: Preliminariesmentioning

confidence: 99%

“…The respective examples exist even with A ∈ M 2 (C) and can be found in [9]; the closedness criterion is given by [18,Theorem 6.4]. Property (g) was proved in [18,Section 3], while the path-connectedness of F N (A) was established earlier in [6,Proposition 7]. Note that, as opposed to F (A), the set F N (A) is not necessarily convex: in particular, for normal A ∈ M 2 (C) it was shown in [8] that F N (A) is a hyperbolic arc.…”

Section: Preliminariesmentioning

confidence: 99%

“…x * Ax x Ax : x ∈ C n , Ax = 0 , it was introduced in [2], and then further investigated in [6]- [10] and [18]. Some of the elementary properties of F N (A) are similar to those of F (A), and can be proved along the same lines.…”

Section: Introductionmentioning

confidence: 99%

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The normalized numerical range and the Davis–Wielandt shell

Lins

Spitkovsky

Zhong

2018

Linear Algebra and its Applications

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For a given n-by-n matrix A, its normalized numerical range F N (A) is defined as the range of the function f N,A : x → (x * Ax)/( Ax · x ) on the complement of ker A. We provide an explicit description of this set for the case when A is normal or n = 2. This extension of earlier results for particular cases of 2-by-2 matrices (by Gevorgyan) and essentially Hermitian matrices of arbitrary size (by A. Stoica and one of the authors) was achieved due to the fresh point of view at F N (A) as the image of the Davis-Wielandt shell DW(A) under a certain non-linear mapping h : R 3 → C.

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Section: Normal Casementioning

confidence: 84%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The normalized numerical range and the Davis–Wielandt shell

Lins

Spitkovsky

Zhong

2018

Linear Algebra and its Applications

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“…Electron positron pair plasma fills a pulsar's magnetosphere, which can be described using force-free electrodynamics Goldreich & Julian (1969). A timedeveloped method can be used to create steady pulsarmagnetosphere solutions, as suggested by Spitkovsky (2006), Komissarov (2006), McKinney (2006). Electric current flows along open magnetic-field lines whereas Poynting flux is radiated outward beyond the light cylinder.…”

Section: Introductionmentioning

confidence: 99%

Global Current Circuit Structure in a Resistive Pulsar Magnetosphere Model

Kato

2017

ApJ

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Pulsar magnetospheres have strong magnetic fields and large amounts of plasma. The structures of these magnetospheres are studied using force-free electrodynamics. To understand pulsar magnetospheres, discussions must include their outer region. However, force-free electrodynamics is limited in it does not handle dissipation. Therefore, a resistive pulsar magnetic field model is needed. To break the ideal magnetohydrodynamic (MHD) condition E · B = 0, Ohm's law is used. In this work, I introduce resistivity depending upon the distance from the star and obtain a self-consistent steady state by time integration. Poloidal current circuits form in the magnetosphere while the toroidal magnetic-field region expands beyond the light cylinder and the Poynting flux radiation appears. High electric resistivity causes a large space scale poloidal current circuit and the magnetosphere radiates a larger Poynting flux than the linear increase outside of the light cylinder radius. The formed poloidal-current circuit has width, which grows with the electric conductivity. This result contributes to a more concrete dissipative pulsar magnetosphere model.

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Systems with strong damping and their spectra

Jacob

Tretter

Trunk

et al. 2018

Math Methods in App Sciences

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We establish a new method for obtaining nonconvex spectral enclosures for operators associated with second-order differential equationsz(t) + Ḋz(t) + A 0 z(t) = 0 in a Hilbert space. In particular, we succeed in establishing the existence of a spectral gap, which is the first result of this kind since the seminal results of Krein and Langer for oscillations of damped systems. While the latter and other spectral bounds are confined to dampings D that are symmetric and dominated by A 0 , we allow for accretive D of equal strength as A 0 .To achieve these results, we prove new abstract spectral inclusion results that are much more powerful than classical numerical range bounds. Two different applications, small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid and wave equations with strong (viscoelastic and frictional) damping, illustrate that our new bounds are explicit. KEYWORDSabstract second-order differential equation, damped system, numerical range, operator matrix, quadratic numerical range, spectrum INTRODUCTIONThe mathematical analysis of abstract second-order Cauchy problems has been a vital field of research over the last decades. In fact, many linear stability problems in applications, particularly in elasticity theory and hydromechanics, are modeled by second-order differential equations of the formin a Hilbert space H, where A 0 is a self-adjoint and uniformly positive operator in H and D is a linear operator in H representing, eg, the damping of the underlying system; see, eg, the works 1-4 ; more recent applications arise in large-scale network dynamic systems; see, eg, the work of Nudell and Chakrabortty. 5 Here, we consider the case that A − 1 2 0 DA − 1 2 0 6546

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On the normalized numerical range

Cited by 7 publications

References 0 publications

The normalized numerical range and the Davis–Wielandt shell

The normalized numerical range and the Davis–Wielandt shell

Global Current Circuit Structure in a Resistive Pulsar Magnetosphere Model

Systems with strong damping and their spectra

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