Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces

Sandovici, Adrian; Snoo, Henk de; Winkler, Henrik

doi:10.1016/j.laa.2007.01.024

Cited by 79 publications

(42 citation statements)

References 14 publications

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“…If no such numbers exist the ascent and descent of A are defined to be ∞. In [13], the authors introduce and study these notions. They showed that many of the results of Taylor and Kaashoek for operators remain valid in the context of linear relations only under the additional condition that the linear relation A has a trivial singular chain manifold, that is, R c (A) = {0}.…”

Section: Algebraic Results For Ascent Essential Ascent Descent and mentioning

confidence: 99%

Ascent, essential ascent, descent and essential descent for a linear relation in a linear space

Chafai

Álvarez

2017

Filomat

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Abstract. For a linear relation in a linear space some spectra defined by means of ascent, essential ascent, descent and essential descent are introduced and studied. We prove that the algebraic ascent, essential ascent, descent and essential descent spectrum of a linear relation in a linear space satisfy the polynomial spectral mapping theorem. As an application of the obtained results we show that the topological ascent, essential ascent, descent and essential descent spectrum verify the polynomial spectral mapping theorem.

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Section: Algebraic Results For Ascent Essential Ascent Descent and mentioning

confidence: 99%

Ascent, essential ascent, descent and essential descent for a linear relation in a linear space

Chafai

Álvarez

2017

Filomat

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“…We adhered to the notations and terminology of the monographs [8,24]. Let E, F and G be linear spaces over K = R or C. A linear relation T from E to F is any mapping having domain D(T ), a nonempty subspace of E, and taking values in the collection of nonempty subsets of F such that T (αx 1 + βx 2 ) = αT (x 1 ) + βT (x 2 ) for all nonzero α, β scalars and x 1 , x 2 ∈ D(T ).…”

Section: Notations and Basic Resultsmentioning

confidence: 99%

Left- and Right-Atkinson Linear Relation Matrices

2015

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Let X and Y be Banach spaces. When A and B are linear relations in X and Y, respectively, we denote by MC the linear relation in X × Y of the form MC = A C 0 B , where 0 is the zero operator from X to Y and C is a bounded operator from Y to X. The goal of this paper is to present some necessary and sufficient conditions on A and B such that there exists a bounded operator C from Y to X for which MC is a Browder linear relation in X × Y.Mathematics Subject Classification. 47A06, 47A53, 47A55.

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“…By [25,Lemma 3.4] and the fact that A is a linear relation in a finite dimensional space C n , there exists a natural number n 0 ≤ n such that ker (…”

Section: It Is Clear That Ker (mentioning

confidence: 99%

Linear relations and the Kronecker canonical form

Berger

Trunk

Winkler

2016

Linear Algebra and its Applications

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Abstract. We show that the Kronecker canonical form (which is a canonical decomposition for pairs of matrices) is the representation of a linear relation in a finite dimensional space. This provides a new geometric view upon the Kronecker canonical form. Each of the four entries of the Kronecker canonical form has a natural meaning for the linear relation which it represents. These four entries represent the Jordan chains at finite eigenvalues, the Jordan chains at infinity, the so-called singular chains and the multi-shift part. Or, to state it more concise: For linear relations the Kronecker canonical form is the analogue of the Jordan canonical form for matrices.

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Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces

Cited by 79 publications

References 14 publications

Ascent, essential ascent, descent and essential descent for a linear relation in a linear space

Ascent, essential ascent, descent and essential descent for a linear relation in a linear space

Left- and Right-Atkinson Linear Relation Matrices

Linear relations and the Kronecker canonical form

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