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.

scholarly journals On boundary value problems for degenerate differential inclusions in Banach spaces

2003 ◽  
Vol 2003 (13) ◽  
pp. 769-784 ◽  
Author(s):  
Valeri Obukhovskii ◽  
Pietro Zecca
Keyword(s):  
Banach Space ◽  
Boundary Value Problems ◽  
Differential Inclusions ◽  
Reflexive Banach Space ◽  
Mild Solutions ◽  
Boundary Value ◽  
Degree Theory ◽  
Linear Operators ◽  
Topological Degree Theory ◽  
Periodic Problems

We consider the applications of the theory of condensing set-valued maps, the theory of set-valued linear operators, and the topological degree theory of the existence of mild solutions for a class of degenerate differential inclusions in a reflexive Banach space. Further, these techniques are used to obtain the solvability of general boundary value problems for a given class of inclusions. Some particular cases including periodic problems are considered.

scholarly journals On linear operators on ordered banach spaces

1983 ◽  
Vol 27 (2) ◽  
pp. 285-305 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Positive Linear Operator ◽  
Linear Operators ◽  
Order Structure ◽  
Ordered Banach Space ◽  
Continuous Linear ◽  
Positive Unit ◽  
Ordered Banach Spaces ◽  
Continuous Linear Operators

The order structure of the space of all continuous linear operators on an ordered Banach space is studied. The main topic is the Robinson property, that is, the norm of a positive linear operator is attained on the positive unit cone.

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.

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.

scholarly journals Absolute values in orthogonally decomposable spaces

1985 ◽  
Vol 31 (2) ◽  
pp. 215-233 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Ordered Banach Space ◽  
Locality Condition ◽  
Absolute Value ◽  
General Properties ◽  
The Absolute ◽  
Kato’S Inequality

In an ordered Banach space which is orthogonally decomposable, we define the absolute value and its general properties are given. The results are used to study the properties of linear operators which satisfy Kato's inequality and the locality condition.

On Spaces of Compact Operators in Non-Archimedean Banach Spaces

1989 ◽  
Vol 32 (4) ◽  
pp. 450-458
Author(s):  
Takemitsu Kiyosawa
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convergent Sequence ◽  
Compact Operators ◽  
Linear Operators ◽  
Continuous Linear ◽  
Infinite Dimensional ◽  
Valued Field ◽  
Infinite Dimensional Banach Space ◽  
Continuous Linear Operators

AbstractLet K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.

Continuous linear operators on C(K, X) and pointwise weakly precompact subsets of C(K, X)

1992 ◽  
Vol 111 (1) ◽  
pp. 143-150 ◽  
Author(s):  
A. lger
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Supremum Norm ◽  
Continuous Linear ◽  
Continuous Linear Operators

AbstractLet K be a compact Hausdorif space, X a Banach space and C(K, X) the Banach space of all continuous functions : KX equipped with the supremum norm. A subset H of C(K, X) is pointwise weakly precompact if, for each t in K, the set Ht) = {(t):H} is weakly precompact. In this note we study the images of a bounded pointwise weakly precompact subset H of C(K, X) under several classes of linear operators on C(K, X).

scholarly journals Common fixed point theorems for a pair of countably condensing mappings in ordered banach spaces

2003 ◽  
Vol 16 (3) ◽  
pp. 243-248 ◽  
Author(s):  
B. C. Dhage ◽  
Donal O'Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Ordered Banach Space ◽  
Isotone Mappings ◽  
Common Fixed Point Theorems ◽  
Ordered Banach Spaces

In this paper some common fixed point theorems for a pair of multivalued weakly isotone mappings on an ordered Banach space are proved.

scholarly journals Commutation Properties of Operator Polynomials

1971 ◽  
Vol 12 (1) ◽  
pp. 98-100
Author(s):  
S. R. Caradus
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Reverse Implication ◽  
Do So ◽  
Continuous Linear Operators

Suppose A and B are continuous linear operators mapping a complex Banach space X into itself. For any polynomial pC, it is obvious that when A commutes with B, then p(A) commutes with B. To see that the reverse implication is false, let A be nilpotent of order n. Then An commutes with all B but A cannot do so. Sufficient conditions for the implication: p(A) commutes with B implies A commutes with B: were given by Embry [2] for the case p(λ) = λn and Finkelstein and Lebow [3] in the general case. The latter authors proved in fact that if f is a function holomorphic on σ(A) and if f is univalent with non-vanishing derivative on σ(A), then A can be expressed as a function of f(A).

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