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.

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 A characterisation and two examples of Riesz operators

1968 ◽  
Vol 9 (2) ◽  
pp. 106-110 ◽  
Author(s):  
T. A. Gillespie ◽  
T. T. West
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Theory ◽  
Bounded Linear Operator ◽  
Spectral Projection ◽  
Resolvent Operators ◽  
Riesz Operator ◽  
Bounded Linear ◽  
Riesz Operators

A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. §3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.

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.

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.

PROPERTY (Bgw) AND PERTURBATIONS

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
M.H.M. Rashid ◽  
T. Prasad
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Point Spectrum ◽  
Space Operator ◽  
Approximate Point Spectrum ◽  
Finite Rank Operators ◽  
Riesz Operators ◽  
Nilpotent Operators ◽  
The Stability ◽  
Isolated Eigenvalues

AbstractA Banach space operator T satisfies property (Bgw) if the complement in the approximate point spectrum σa(T) of the semi-B-essential approximate point spectrum σSHF+-(T) coincides with the set of isolated eigenvalues of T of Unite multiplicity E°(T). We find conditions for Banach Space operator tosatfafy the property (Bgw). We also study the stability of property (Bgw) under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators and by Riesz operators commuting with T.

scholarly journals A direct proof of a theorem of West on sequences of Riesz operators

1974 ◽  
Vol 15 (2) ◽  
pp. 93-94
Author(s):  
Anthony F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Direct Proof ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Riesz Operator ◽  
Bounded Linear ◽  
Riesz Operators

We recall (cf. [2] Definitions 3.1 and 3.2, p. 322) that a bounded linear operator T on a Banach space ℵ into itself is said to be asymptotically quasi-compact if K(Tn)⅟n → 0 as n → ∞. where K(U) = inf ∥U–C∥ for every bounded linear operator U on ℵ into itself, the infimum being taken over all compact linear operators C on ℵ into itself. For a complex Banach space, this is equivalent (cf. [2], pp. 319, 321 and 326) to T being a Riesz operator.

A contribution to the solution of the compact correction problem for operators on a Banach space

1989 ◽  
Vol 31 (2) ◽  
pp. 219-229
Author(s):  
Mícheál Ó Searcóid
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Compact Operator ◽  
Complex Number ◽  
Space Problem ◽  
Riesz Operators ◽  
Correction Problem ◽  
Bounded Below ◽  
General Banach Space

We consider the hypothesis that an operator T on a given Banach space can always be perturbed by a compact operator K in such a way that, whenever a complex number A is in the semi-Fredholm region of T + K, then T + K – λ is either bounded below or surjective. The hypothesis has its origin in the work of West [11], who proved it for Riesz operators on Hilbert space. In this paper, we reduce the general Banach space problem to one of considering only operators of a special type, operators which are, in a spectral sense, natural generalizations of the Riesz operators studied by West.

scholarly journals The evaluation functionals associated with an algebra of bounded operators

1969 ◽  
Vol 10 (1) ◽  
pp. 73-76 ◽  
Author(s):  
J. Duncan
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Finite Rank ◽  
Bounded Operators ◽  
Image Position

In this note we shall employ the notation of [1] without further mention. Thus X denotes a normed space and P the subset of X × X′ given byGiven a subalgebra of B(X), the set {Φ(X,f):(x,f) ∈ P} of evaluation functional on is denoted by II. We shall prove that if X is a Banach space and if contains all the bounded operators of finite rank, then Π is norm closed in ′. We give an example to show that Π need not be weak* closed in ″. We show also that FT need not be norm closed in ″ if X is not complete.

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