Left-right fredholm and left-right browder linear relations

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 255-271 ◽  
Author(s):  
T. Álvarez ◽  
Fatma Fakhfakh ◽  
Maher Mnif
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Linear Relation ◽  
Finite Rank Operator ◽  
Linear Relations ◽  
Rank Operator ◽  
Type Decomposition ◽  
Converse Results

In this paper we introduce the notions of left (resp. right) Fredholm and left (resp. right) Browder linear relations. We construct a Kato-type decomposition of such linear relations. The results are then applied to give another decomposition of a left (resp. right) Browder linear relation T in a Banach space as an operator-like sum T = A + B, where A is an injective left (resp. a surjective right) Fredholm linear relation and B is a bounded finite rank operator with certain properties of commutativity. The converse results remain valid with certain conditions of commutativity. As a consequence, we infer the characterization of left (resp. right) Browder spectrum under finite rank operator.

scholarly journals The Local spectral theory and surjective spectrum of linear relations

2021 ◽  
Vol 73 (2) ◽  
pp. 222-237
Author(s):  
M. Mnif ◽  
A.-A. Ouled-Hmed
Keyword(s):  
Banach Space ◽  
Spectral Theory ◽  
Linear Relation ◽  
Local Spectrum ◽  
Linear Relations ◽  
Analytic Core ◽  
Local Spectral Theory

UDC 517.98 This paper initiates a study of local spectral theory for linear relations. At the beginning, we define the local spectrum and study its properties. Then we obtain results related to the correlation analytic core and quasinilpotent part of a linear relation in a Banach space . As an application, we give a characterization of the surjective spectrum in terms of the local spectrum and show that if , then does not cluster at .

scholarly journals FINITE RANK RIESZ OPERATORS

2013 ◽  
Vol 56 (1) ◽  
pp. 183-185 ◽  
Author(s):  
U. KOUMBA ◽  
H. RAUBENHEIMER
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Riesz Operator ◽  
Finite Rank Operator ◽  
Rank Operator ◽  
Riesz Operators

AbstractWe provide conditions under which a Riesz operator defined on a Banach space is a finite rank operator.

scholarly journals On strictly quasi-Fredholm linear relations and semi-B-Fredholm linear relation perturbations

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6337-6355 ◽  
Author(s):  
Bouaniza Hafsa ◽  
Maher Mnif
Keyword(s):  
Finite Rank ◽  
Linear Relation ◽  
Linear Relations ◽  
Finite Rank Operators ◽  
Lower Semi ◽  
The Stability

In this paper we introduce the set of strictly quasi-Fredholm linear relations and we give some of its properties. Furthermore, we study the connection between this set and some classes of linear relations related to the notions of ascent, essentially ascent, descent and essentially descent. The obtained results are used to study the stability of upper semi-B-Fredholm and lower semi-B-Fredholm linear relations under perturbation by finite rank operators.

scholarly journals Generalized Kato linear relations

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1129-1139 ◽  
Author(s):  
Mohammed Benharrat ◽  
Teresa Álvarez ◽  
Bekkai Messirdi
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Relation ◽  
Shift Operator ◽  
Linear Operators ◽  
Symmetric Difference ◽  
Linear Relations ◽  
Left Shift ◽  
Kato Spectrum

For a Banach space the notions of generalized Kato linear relation and the corresponding spectrum are introduced and studied. We show that the symmetric difference between the generalized Kato spectrum and the Goldberg spectrum of multivalued linear operators in Banach spaces is at most countable. The obtained results are used to describe the generalized Kato spectrum of the inverse of the left shift operator regarded as a linear relation.

scholarly journals Some fundamental properties of fuzzy linear relations between vector spaces

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 41-53 ◽  
Author(s):  
Sorin Nădăbana
Keyword(s):  
Linear Relation ◽  
Scalar Multiplication ◽  
Vector Spaces ◽  
Fuzzy Relations ◽  
Linear Relations ◽  
Fundamental Properties ◽  
The Family ◽  
Natural Way

This paper aims at studying the fundamental properties of fuzzy linear relations between vector spaces. The sum of two fuzzy relations and the scalar multiplication are defined, in a natural way, and some properties of this operations are established. Fuzzy linear relations are investigated and among the results obtained, there should be underlined a characterization of fuzzy linear relations and the fact that the inverse of a fuzzy linear relation is also a fuzzy linear relation. Moreover, the paper shows that the composition of two fuzzy linear relations is a fuzzy linear relation as well. Finally, the article highlights that the family of all fuzzy linear relations is closed under addition and it is closed under scalar multiplication.

scholarly journals On the stability of the spectral properties under commuting perturbations

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1511-1518
Author(s):  
Kai Yan ◽  
Weigang Su ◽  
Xiaochun Fang
Keyword(s):  
Finite Rank ◽  
Spectral Properties ◽  
Finite Rank Operator ◽  
Rank Operator ◽  
Algebraic Operator ◽  
The Stability

In this paper, we examine the stability of several spectral properties under commuting perturbations. In particular, we show that if T ( L(X) is an isoloid operator satisfying generalized Weyl?s theorem and if F ( L(X) is a power finite rank operator that commutes with T, then generalized Weyl?s theorem holds for T + F. In addition, we consider the permanence of Bishop?s property (?), at a point, under commuting perturbation that is an algebraic operator.

Spectral characterization of the socle in Jordan–Banach algebras

1995 ◽  
Vol 117 (3) ◽  
pp. 479-489 ◽  
Author(s):  
Bernard Aupetit
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Finite Rank ◽  
Banach Algebras ◽  
Spectral Characterization ◽  
Bounded Operators ◽  
Finite Rank Operators ◽  
Finite Dimensional ◽  
Minimal Idempotent

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

scholarly journals Decomposability of finite rank operators in certain subspaces and algebras

2001 ◽  
Vol 64 (2) ◽  
pp. 307-314
Author(s):  
Jiankui Li
Keyword(s):  
Hilbert Space ◽  
Finite Rank ◽  
Bounded Operators ◽  
Subspace Lattice ◽  
Finite Rank Operator ◽  
Rank Operator ◽  
Finite Rank Operators ◽  
Reflexive Algebra ◽  
Reflexive Subspace ◽  
Rank One

Let  be either a reflexive subspace or a bimodule of a reflexive algebra in B (H), the set of bounded operators on a Hilbert space H. We find some conditions such that a finite rank T ∈  has a rank one summand in  and  has strong decomposability. Let (ℒ) be the set of all operators on H that annihilate all the operators of rank at most one in alg ℒ. We construct an atomic Boolean subspace lattice ℒ on H such that there is a finite rank operator T in (ℒ) such that T does not have a rank one summand in (ℒ). We obtain some lattice-theoretic conditions on a subspace lattice ℒ which imply alg ℒ is strongly decomposable.

Sign in / Sign up

Export Citation Format

Share Document