Invariance of the Essential Spectra of Operator Pencils

Gernandt, Hannes; Moalla, Nedra; Philipp, Friedrich; Selmi, Wafa; Trunk, Carsten

doi:10.1007/978-3-030-43380-2_10

Cited by 5 publications

(4 citation statements)

References 14 publications

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“…There is a deep connection between the range and the kernel representation and the corresponding matrix pencil. This was already utilized in [8,14,15,16,21]. A complete set of invariants for the Kronecker canonical form are four multi-indices: the finite and infinite elementary divisors, the column and the row minimal indices.…”

Section: Main Result: Jordan-like Decompositionmentioning

confidence: 99%

A Jordan-like decomposition for linear relations in finite-dimensional spaces

Berger¹,

Snoo²,

Trunk³

et al. 2022

Preprint

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A square matrix A has the usual Jordan canonical form that describes the structure of A via eigenvalues and the corresponding Jordan blocks. If A is a linear relation in a finite-dimensional linear space H (i.e., A is a linear subspace of H × H and can be considered as a multivalued linear operator), then there is a richer structure. In addition to the classical Jordan chains (interpreted in the Cartesian product H × H), there occur three more classes of chains: chains starting at zero (the chains for the eigenvalue infinity), chains starting at zero and also ending at zero (the singular chains), and chains with linearly independent entries (the shift chains). These four types of chains give rise to a direct sum decomposition (a Jordan-like decomposition) of the linear relation A. In this decomposition there is a completely singular part that has the extended complex plane as eigenvalues; a usual Jordan part that corresponds to the finite proper eigenvalues; a Jordan part that corresponds to the eigenvalue ∞; and a multishift, i.e., a part that has no eigenvalues at all. Furthermore, the Jordan-like decomposition exhibits a certain uniqueness, closing a gap in earlier results. The presentation is purely algebraic, only the structure of linear spaces is used. Moreover, the presentation has a uniform character: each of the above types is constructed via an appropriately chosen sequence of quotient spaces. The dimensions of the spaces are the Weyr characteristics, which uniquely determine the Jordan-like decomposition of the linear relation.

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Section: Main Result: Jordan-like Decompositionmentioning

confidence: 99%

A Jordan-like decomposition for linear relations in finite-dimensional spaces

Berger¹,

Snoo²,

Trunk³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Below, we describe the relation between the spectrum of operator pencils and the associated linear relations. For related results see [24,34].…”

Section: Linear Relationsmentioning

confidence: 99%

“…(19), is finite. As A is injective on ker E ∩ dom A (see (24) above), also ker E ∩ dom A is of finite dimension and E| dom A is Fredholm. Hence, also (A − µE) −1 E| dom A is Fredholm and in particular, by (18), dom (E| −1…”

Section: Linear Relationsmentioning

confidence: 99%

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The spectrum and the Weyr characteristics of operator pencils and linear relations

Gernandt,

Trunk

2021

Preprint

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The relation between the spectra of operator pencils with unbounded coefficients and of associated linear relations is investigated. It turns out that various types of spectrum coincide and the same is true for the Weyr characteristics. This characteristic describes how many independent Jordan chains up to a certain length exist. Furthermore, the change of this characteristic subject to one-dimensional perturbations is investigated.

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On the Essential Spectrum of Operator Pencils

Wilson,

Khelif,

Trunk

2024

Proc Appl Math and Mech

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For a closed densely defined linear operator and a bounded linear operator on a Banach space whose essential spectrums are contained in disjoint sectors, we show that the essential spectrum of the associated operator pencil is contained in a sector of the complex plane whose boundaries are determined purely by the angles that define the two sectors, which contain the essential spectrums of and .

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Invariance of the Essential Spectra of Operator Pencils

Cited by 5 publications

References 14 publications

A Jordan-like decomposition for linear relations in finite-dimensional spaces

A Jordan-like decomposition for linear relations in finite-dimensional spaces

The spectrum and the Weyr characteristics of operator pencils and linear relations

On the Essential Spectrum of Operator Pencils

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