scholarly journals The local spectral radius of a nonnegative orbit of compact linear operators

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Mihály Pituk
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Additional Assumption ◽  
Real Banach Space ◽  
Linear Operators ◽  
Partial Ordering ◽  
Positive Eigenvector ◽  
Local Spectral Radius

AbstractWe consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.

scholarly journals Essential supremum norm differentiability

1985 ◽  
Vol 8 (3) ◽  
pp. 433-439
Author(s):  
I. E. Leonard ◽  
K. F. Taylor
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Real Banach Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Supremum Norm ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Fréchet Differentiability ◽  
Finite Measure Space

The points of Gateaux and Fréchet differentiability inL∞(μ,X)are obtained, where(Ω,∑,μ)is a finite measure space andXis a real Banach space. An application of these results is given to the spaceB(L1(μ,ℝ),X)of all bounded linear operators fromL1(μ,ℝ)intoX.

scholarly journals Stronger maximal monotonicity properties of linear operators

1999 ◽  
Vol 60 (1) ◽  
pp. 163-174 ◽  
Author(s):  
H.H. Bauschke ◽  
S. Simons
Keyword(s):  
Banach Space ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Real Banach Space ◽  
Lower Semicontinuous ◽  
Maximal Monotone ◽  
Linear Mapping ◽  
Linear Operators ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function

The subdifferential mapping associated with a proper, convex lower semicontinuous function on a real Banach space is always a special kind of maximal monotone operator. Specifically, it is always “strongly maximal monotone” and of “type (ANA)”. In an attempt to find maximal monotone operators that do not satisfy these properties, we investigate (possibly discontinuous) maximal monotone linear operators from a subspace of a (possibly nonreflexive) real Banach space into its dual. Such a linear mapping is always “strongly maximal monotone”, but we are only able to prove that is of “type (ANA)” when it is continuous or surjective — the situation in general is unclear. In fact, every surjective linear maximal monotone operator is of “type (NA)”, a more restrictive condition than “type (ANA)”, while the zero operator, which is both continuous and linear and also a subdifferential, is never of “type (NA)” if the underlying space is not reflexive. We examine some examples based on the properties of derivatives.

Non-compact positive operators

1972 ◽  
Vol 328 (1572) ◽  
pp. 67-81 ◽  
Keyword(s):  
Banach Space ◽  
Fixed Point Theorems ◽  
Degree Theory ◽  
Positive Operators ◽  
Linear Operators ◽  
Ordered Banach Space ◽  
Continuous Linear ◽  
Topological Degree Theory ◽  
Positive Eigenvector ◽  
Natural Way

This paper is concerned with positive operators acting in a partially ordered Banach space, and provides extensions of various theorems which involve operators of this kind which are in addition supposed to be compact. It is shown that any continuous linear positive map T has a positive eigenvector provided the spectrum of T contains a point with modulus greater than the radius of the essential spectrum of T : this result contains the well-known theorem of Krein-Rutman for compact operators. Various results connected to the Krein-Rutman theorem in a natural way are provided for non-compact positive linear operators, some involving k -set contractions and another which utilizes the notion of a projectionally compact operator. Two fixed point theorems for nonlinear positive operators are obtained by the use of topological degree theory for k -set contractions.

Nonlinear preservers of group invertible operators

2017 ◽  
Vol 11 (01) ◽  
pp. 1850002 ◽  
Author(s):  
M. Oudghiri ◽  
K. Souilah
Keyword(s):  
Banach Space ◽  
Real Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Nonlinear Preservers ◽  
The Difference ◽  
Nonzero Scalar

Let [Formula: see text] be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space [Formula: see text]. We prove that a bijective bicontinuous map [Formula: see text] on [Formula: see text] preserves the difference of group invertible operators in both directions if and only if [Formula: see text] is either of the form [Formula: see text] or of the form [Formula: see text], where [Formula: see text] is a nonzero scalar, [Formula: see text] and [Formula: see text] are two bounded invertible linear or conjugate linear operators.

scholarly journals Supremum norm differentiability

1983 ◽  
Vol 6 (4) ◽  
pp. 705-713
Author(s):  
I. E. Leonard ◽  
K. F. Taylor
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Real Banach Space ◽  
Linear Operators ◽  
Supremum Norm ◽  
Locally Compact ◽  
Space Applications ◽  
Bounded Linear ◽  
Locally Compact Hausdorff Space ◽  
Bounded Sequences

The points of Gateaux and Fréchet differentiability of the norm inC(T,E)are obtained, whereTis a locally compact Hausdorff space andEis a real Banach space. Applications of these results are given to the spaceℓ∞(E)of all bounded sequences inEand to the spaceB(ℓ1,E)of all bounded linear operators fromℓ1intoE

scholarly journals Spectral radius of S-essential spectra

2020 ◽  
Vol 54 (1) ◽  
pp. 91-97
Author(s):  
C. Belabbaci
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Essential Spectrum ◽  
Measure Of Noncompactness ◽  
Dimensional Subspace ◽  
Linear Operators ◽  
Essential Spectra ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Kuratowski Measure Of Noncompactness

In this paper, we study the spectral radius of some S-essential spectra of a bounded linear operator defined on a Banach space. More precisely, via the concept of measure of noncompactness,we show that for any two bounded linear operators $T$ and $S$ with $S$ non zero and non compact operator the spectral radius of the S-Gustafson, S-Weidmann, S-Kato and S-Wolf essential spectra are given by the following inequalities\begin{equation}\dfrac{\beta(T)}{\alpha(S)}\leq r_{e, S}(T)\leq \dfrac{\alpha(T)}{\beta(S)},\end{equation}where $\alpha(.)$ stands for the Kuratowski measure of noncompactness and $\beta(.)$ is defined in [11].In the particular case when the index of the operator $S$ is equal to zero, we prove the last inequalities for the spectral radius of the S-Schechter essential spectrum. Also, we prove that the spectral radius of the S-Jeribi essential spectrum satisfies inequalities 2) when the Banach space $X$ has no reflexive infinite dimensional subspace and the index of the operator $S$ is equal to zero (the S-Jeribi essential spectrum, introduced in [7]as a generalisation of the Jeribi essential spectrum).

scholarly journals Properties of Chmielinski-orthogonality using Kadets-Klee property

2022 ◽  
pp. 94-101
Author(s):  
saied Johnny ◽  
Buthainah A. A. Ahmed
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Real Banach Space ◽  
Convex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
James Orthogonality ◽  
New Type

The aim of this paper is to study new results of an approximate orthogonality of Birkhoff-James techniques in real Banach space , namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by orthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. Also, we explore relation between two different types of orthogonalities. First of them orthogonality in a real Banach space and the other orthogonality in the space of bounded linear operator . We obtain a complete characterizations of these two orthogonalities in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. The study is designed to give different results about the concept symmetry of Chmielinski-orthogonality for a compact linear operator on a reflexive, strictly convex Banach space having Kadets-Klee property by exploring a new type of a generalized some results with Birkhoff James orthogonality in the space of bounded linear operators. We also exhibit a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having Kadets-Klee property either having zero nullity or not -right-symmetric.

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

An operator version of Abel's continuity theorem

2010 ◽  
Vol 17 (4) ◽  
pp. 787-794
Author(s):  
Vaja Tarieladze
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Identity Operator ◽  
Continuous Linear ◽  
Continuity Theorem ◽  
Operator Version ◽  
Continuous Linear Operators

Abstract For a Banach space X let 𝔄 be the set of continuous linear operators A : X → X with ‖A‖ < 1, I be the identity operator and 𝔄 c ≔ {A ∈ 𝔄 : ‖I – A‖ ≤ c(1 – ‖A‖)}, where c ≥ 1 is a constant. Let, moreover, (xk ) k≥0 be a sequence in X such that the series converges and ƒ : 𝔄 ∪ {I} → X be the mapping defined by the equality It is shown that ƒ is continuous on 𝔄 and for every c ≥ 1 the restriction of ƒ to 𝔄 c ∪ {I} is continuous at I.

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