scholarly journals Sufficient conditions for a continuous linear operator to be weakly compact

1972 ◽  
Vol 7 (2) ◽  
pp. 183-190 ◽  
Author(s):  
Joe Howard ◽  
Kenneth Melendez
Keyword(s):  
Linear Operator ◽  
Sufficient Conditions ◽  
Continuous Linear Operator ◽  
Continuous Operator ◽  
Continuous Linear ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Continuous Operator ◽  
Locally Convex

A locally convex topological vector (LCTV) space E is said to have property V (Dieudonné property) if for every complete separated LCTV space F, every unconditionally converging (weakly completely continuous) operator T: E → F is wsakly compact. First, an investigation of the permanence of property V is given. The permanence of the Dieudonné is analogous. Relationships between property V and the Dieudonné property are then given.

scholarly journals Superdiagonal forms for completely continuous linear operators

1982 ◽  
Vol 23 (2) ◽  
pp. 163-170 ◽  
Author(s):  
Demetrios Koros
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Topological Vector Space ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Complex Field ◽  
Continuous Linear ◽  
Completely Continuous ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex

Altman [1] showed that Riesz-Schauder theory remains valid for a completely continuous linear operator on a locally convex Hausdorflf topological vector space over the complex field. In a later paper [2], he proved an analogue of the Aronszajn-Smith result; specifically, he showed that such an operator possesses a proper closed invariant subspace. The purpose of this paper is to show that Ringrose's theory of superdiagonal forms for compact linear operators [3] can be generalized to the case of a completely continuous linear operator on a locally convex Hausdorff topological vector space over the complex field. However, the proof given in [3] requires considerable modification.

scholarly journals ON A CLASS OF POLYA PROPERTY PRESERVING OPERATORS

2020 ◽  
Vol 27 (4) ◽  
pp. 301-313
Author(s):  
CHIU-CHENG CHANG
Keyword(s):  
Linear Operator ◽  
Integral Representation ◽  
Kernel Function ◽  
Sufficient Conditions ◽  
Continuous Linear Operator ◽  
Continuous Linear

In this paper, we show that every continuous linear operator from H(OmegawXOmegaz) to H (OmegawxOmegaxi) has an integral representation with a kernel function M(z, w, xi). We give two sufficient conditions on M(z,w,() to ensure that its corresponding operator preserves Polya property. We also prove that a continuous linear operator from H(fl,,, x ) to H(! x S2() either preserves the Polya property for all functions with that property or does not preserve the Polya property for any function.

scholarly journals Spectral operators and weakly compact homomorphisms in a class of Banach Spaces

1986 ◽  
Vol 28 (2) ◽  
pp. 215-222 ◽  
Author(s):  
W. Ricker
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Finite Type ◽  
Continuous Linear Operator ◽  
Multiplication Operators ◽  
Continuous Linear ◽  
Simple Application ◽  
Spectral Operator ◽  
Weakly Compact ◽  
Finite Spectrum

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

scholarly journals Pseudospectra in a non-Archimedean Banach space and essential pseudospectra in Eω

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3961-3976
Author(s):  
Aymen Ammar ◽  
Ameni Bouchekoua ◽  
Aref Jeribi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Continuous Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Completely Continuous ◽  
Completely Continuous Operator

In this work, we introduce and study the pseudospectra and the essential pseudospectra of linear operators in a non-Archimedean Banach space and in the non-Archimedean Hilbert space E?, respectively. In particular, we characterize these pseudospectra. Furthermore, inspired by T. Diagana and F. Ramaroson [12], we establish a relationship between the essential pseudospectrum of a closed linear operator and the essential pseudospectrum of this closed linear operator perturbed by completely continuous operator in the non-Archimedean Hilbert space E?.

On eigenvalues and inverse singular values of compact linear operators in Hilbert space

1953 ◽  
Vol 49 (2) ◽  
pp. 201-212 ◽  
Author(s):  
J. P. O. Silberstein ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Continuous Linear Operator ◽  
Singular Values ◽  
Linear Operators ◽  
Complex Numbers ◽  
Identity Operator ◽  
Continuous Linear ◽  
Completely Continuous ◽  
Image Position

1·1. In this paper we shall be concerned with the equationswhere K is a compact (completely continuous) linear operator in a Hilbert space , K is the adjoint of K, I is the identity operator, x and y are elements of ∥ x ∥ denotes the norm of x, and κ and σ are complex numbers.

scholarly journals OPERATORS ON C0(L,X) WHOSE RANGE DOES NOT CONTAIN c0

2008 ◽  
Vol 77 (3) ◽  
pp. 515-520
Author(s):  
JARNO TALPONEN
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Continuous Linear Operator ◽  
Continuous Linear ◽  
Locally Compact ◽  
Weakly Compact ◽  
Isolated Points ◽  
Locally Compact Hausdorff Space

AbstractThis paper contains two results: (a) if $\mathrm {X}\neq \{0\}$ is a Banach space and (L,τ) is a nonempty locally compact Hausdorff space without isolated points, then each linear operator T:C0(L,X)→C0(L,X) whose range does not contain an isomorphic copy of c00 satisfies the Daugavet equality $\|\mathbf {I}+T\|=1+\|T\|$; (b) if Γ is a nonempty set and X and Y are Banach spaces such that X is reflexive and Y does not contain c0 isomorphically, then any continuous linear operator T:c0(Γ,X)→Y is weakly compact.

scholarly journals Linear operators whose domain is locally convex

1977 ◽  
Vol 20 (4) ◽  
pp. 293-299 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Topological Vector Space ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Continuous Linear ◽  
Locally Convex

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on S separates the points of S. If X is a Banach space and T : X → F is a continuous linear operator, then T is quasi-convex if is quasi-convex, where U is the unit ball of X.

scholarly journals Another proof of a result of N. J. Kalton, E. Saab and P. Saab on the Dieudonné property in C(K, E)

1989 ◽  
Vol 31 (2) ◽  
pp. 137-140 ◽  
Author(s):  
G. Emmanuele
Keyword(s):  
Vector Measure ◽  
Measure Theory ◽  
Short Note ◽  
Continuous Operator ◽  
Weak Compactness ◽  
Multivalued Mappings ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Continuous Operator ◽  
Integrable Functions

Let K be a compact Hausdorff topological space and E be a Banach space not containing l1. Recently N. J. Kalton, E. Saab and P. Saab ([5]) obtained the results that under the above assumptions the usual space C(K, E) has the Dieudonné property; i.e. each weakly completely continuous operator on C(K, E) is weakly compact. They use topological results concerning multivalued mappings in their proof. In this short note we furnish a new and simpler proof of that result without using topological results but only well known theorems of Bourgain ([2]) and Talagrand ([8]) on weak compactness of sets of Bochner integrable functions; i.e. results in vector measure theory. At the end of the paper we present some applications of the result to Banach spaces of compact operators.

scholarly journals Semi-embeddings of Banach spaces which are hereditarily c0

1983 ◽  
Vol 26 (2) ◽  
pp. 163-167 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Closed Unit Ball ◽  
Continuous Linear ◽  
Scattered Space ◽  
Complete Set

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

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