Pseudospectrum Enclosures by Discretization

Frommer, Andreas; Jacob, Birgit; Vorberg, Lukas A.; Wyss, Christian; Zwaan, Ian N.

doi:10.1007/s00020-020-02621-5

Cited by 4 publications

(2 citation statements)

References 20 publications

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“…But, while these sequences of inclusion sets are of significant theoretical interest and converge in many cases to Spec A, it is unclear, for general operators A and particularly for larger n, how to realise these sets computationally. (However, see the related work of Frommer et al [43] on computation of inclusion sets for pseudospectra via approximate computation of numerical ranges of resolvents of the operator, and work of Bögli and Marletta [6] on inclusion sets for spectra via numerical ranges of related operator pencils. )…”

Section: Related Workmentioning

confidence: 99%

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

Chandler-Wilde,

Chonchaiya,

Lindner

2024

J. Spectr. Theory

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In this paper, we derive novel families of inclusion sets for the spectra and pseudospectra of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case, we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one in the sense of Ben-Artzi et al. (2020). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies [Commun. Math. Phys. 216 (2001), 687–704].

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Section: Related Workmentioning

confidence: 99%

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

Chandler-Wilde,

Chonchaiya,

Lindner

2024

J. Spectr. Theory

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“…We refer to Landau [17,18], Trefthen [26], Böttcher [5,6], Davies [22] for some classical work on pseudospectra. Also see [7,8,9,13,19,20,28] for recent developments.…”

Section: Introductionmentioning

confidence: 99%

Pseudo $S$-spectra of special operators in quaternionic Hilbert spaces

Dhara¹,

Pamula²

2022

Preprint

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For a bounded quaternionic operator T on a right quaternionic Hilbert space H and ε > 0, the pseudo S-spectrum of T is defined aswhere H denotes the division ring of quaternions, σ S (T ) is the S-spectrum of T and ∆ q (T ) = T 2 − 2Re(q)T + |q| 2 I. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo S-spectrum and explicitly compute the pseudo S-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of S-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number r if and only if for every ε > 0 the pseudo Sspectrum of the operator is the circularization of a closed disc in the complex plane centered at r with the radius √ ε. Further, we propose a G 1 -condition for quaternionic operators and prove some results in this setting.

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Spectral Approximation of Generalized Schrödinger Operators via Approximation of Subwords

Gabel,

Gallaun,

Grossmann

et al. 2023

Complex Anal. Oper. Theory

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We prove criteria, purely based on finite subwords of the potential, for spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schrödinger operators on the discrete line or half-line. In fact, our results are neither limited to Schrödinger or self-adjoint operators, nor to Hilbert space or 1D: By employing localized lower norms, we strongly generalize known results from the self-adjoint case to much more general and non-normal situations, including various configurations of Hamiltonians and further non-self-adjoint models with aperiodic or pseudoergodic potentials, even models with multiple varying diagonals and entries in a Banach space.

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Pseudospectrum Enclosures by Discretization

Cited by 4 publications

References 20 publications

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

Pseudo $S$-spectra of special operators in quaternionic Hilbert spaces

Spectral Approximation of Generalized Schrödinger Operators via Approximation of Subwords

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