Perturbation Results for Multivalued Linear Operators

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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“…Arguing exactly as in the proof of Theorem VI.5.4 in [8] we obtain that ρ((A − λ i ) m i ) ∅ for each i ∈ {1, 2, ..., p} and that ρ(P(A)) ∅.…”

Section: Definition 32mentioning

confidence: 59%

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

Chafai

Álvarez

2017

Filomat

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“…The desired conclusion now follows upon noting that k(T) = −k(T ) and k(T + S) = −k((T + S) ) by virtue of [7,Proposition V.15.3]. In this situation since T is closed, we deduce from the part (i) applied to T and S that T + S is a closed F + -relation and k(T ) = k(T + S ).…”

Section: (I) T Is Not Upper Semi-fredholm If and Only If There Is No mentioning

confidence: 92%

“…We recall some basic definitions and properties of linear relations in normed spaces following the notation and terminology of the book [7]. Let X, Y and Z denote infinite dimensional normed spaces over ‫ދ‬ = ‫ޒ‬ or ‫.ރ‬ A linear relation or multi-valued linear operator T from X to Y is a mapping from a subspace D(T) = {x ∈ X : Tx = ∅}, called the domain of T, into the collection of non-empty subsets of Y such that T(αx 1 + βx 2 ) = αTx 1 + βTx 2 for all non-zero α, β scalars and x 1 , x 2 ∈ D(T).…”

Section: Introductionmentioning

confidence: 99%

“…">Introduction.We recall some basic definitions and properties of linear relations in normed spaces following the notation and terminology of the book [7]. We apply this perturbation result to investigate the stability of almost semi-Fredholm multi-valued linear operators in normed spaces under strictly singular perturbation as well as the behaviour of the index under perturbation.…”

mentioning

confidence: 99%

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Strictly Singular Perturbation of Almost Semi-Fredholm Linear Relations in Normed Spaces

Álvarez

2013

Glasgow Math. J.

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In this paper, we introduce the notions of almost upper semi-Fredholm and strictly singular pairs of subspaces and show that the class of almost upper semi-Fredholm pairs of subspaces is stable under strictly singular pairs perturbation. We apply this perturbation result to investigate the stability of almost semi-Fredholm multi-valued linear operators in normed spaces under strictly singular perturbation as well as the behaviour of the index under perturbation.2000 Mathematics Subject Classification. 47A06. Introduction.We recall some basic definitions and properties of linear relations in normed spaces following the notation and terminology of the book [7]. Let X, Y and Z denote infinite dimensional normed spaces over ‫ދ‬ = ‫ޒ‬ or ‫.ރ‬ A linear relation or multi-valued linear operator T from X to Y is a mapping from a subspace D(T) = {x ∈ X : Tx = ∅}, called the domain of T, into the collection of non-empty subsets of Y such that T(αx 1 + βx 2 ) = αTx 1 + βTx 2 for all non-zero α, β scalars and x 1 , x 2 ∈ D(T). The class of such linear relations is denoted by LR(X, Y ). If T maps the points of its domain to singletons, then T is said to be a single valued or simply an operator. A linear relation T is uniquely determined by its graph, G(T), which is defined byY ). The inverse of T is the linear relation T −1 given by G(T −1 ) := {(y, x) : (x, y) ∈ G(T)}. The subspaces T −1 (0) := N(T) and R(T) := T(D(T)) are called the null space and the range of T, respectively. We say that T is injective if N(T) = {0}. We note that D(T −1 ) = R(T), D(T) = R(T −1 ), T is single valued if and only if the subspace T(0) coincides with {0} and y ∈ Tx if and only if Tx = y + T(0). We write α(T) := dimN(T), β(T) := dimY/R(T), β(T) := dimY/R(T), k(T) := α(T) − β(T), provided α(T) and β(T) are not both infinite, and the topological index of T is the quantity α(T) − β(T), provided both α(T) and β(T) are not infinite. For S, T ∈ LR(X, Y ) and R ∈ LR(Y, Z) the sum S + T and the product or composition RS are defined by G(S + T) := {(x, y 1 + y 2 ) : (x, y 1 ) ∈ G(S), (x, y 2 ) ∈ G(T)} and G(RS) := {(x, z) : (x, y) ∈ G(S), (y, z) ∈ G(R) for some y ∈ Y }. Let M and N be subspaces of X and the dual space X respectively. Then J M denotes the natural injection map of M into X, M ⊥ = {x ∈ X : x (M) = 0}, and if M ∩ D(T) = ∅, then T | M is given by G(T | M ) := {(x, y) ∈ G(T) : x ∈ M} and N := {x ∈ X : N(x) = 0}. We observe that T | M = TJ M but T | M = TJ M if T is single valued.

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“…However, it seems that the behavior of the domain, the range, the kernel and the multivalued part of p(S) has not yet been described, cf. [3]. It is the goal of this technical note to cover this gap.…”

Section: Introductionmentioning

confidence: 96%

Some Basic Properties of Polynomials in a Linear Relation in Linear Spaces

Sandovici

Operator Theory in Inner Product Spaces

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Perturbation Results for Multivalued Linear Operators

Cited by 17 publications

References 7 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

Strictly Singular Perturbation of Almost Semi-Fredholm Linear Relations in Normed Spaces

Some Basic Properties of Polynomials in a Linear Relation in Linear Spaces

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