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)).

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 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.

Linearization of a Contractive Homeomorphism

1968 ◽  
Vol 20 ◽  
pp. 1387-1390
Author(s):  
Ludvik Janos
Keyword(s):  
Linear Operator ◽  
Topological Space ◽  
Vector Space ◽  
Topological Vector Space ◽  
Continuous Linear Operator ◽  
Continuous Linear ◽  
Topological Embedding

Let X be a topological space and ϕ: X ⟶ X a continuous self-mapping of X. We say that ϕ is linearized in L by Φ if there exists a topological embedding μ: X ⟶ L of the space X into the linear topological vector space L such that for all x ϵ X, μ (ϕ (x)) = Φ (μ (x)), where ϕ is a continuous linear operator on L.

scholarly journals On the spectrum of ψ-contracting operators

2001 ◽  
Vol 14 (3) ◽  
pp. 303-308 ◽  
Author(s):  
Anwar A. Al-Nayef
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Continuous Linear Operator ◽  
Continuous Linear ◽  
Hausdorff Measure Of Noncompactness

The spectrum σ(A) of a continuous linear operator A:E→E defined on a Banach space E, which is contracting with respect to the Hausdorff measure of noncompactness, is investigated.

scholarly journals On totally paranormal operators

2002 ◽  
Vol 66 (3) ◽  
pp. 425-441 ◽  
Author(s):  
Christoph Schmoeger
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Continuous Linear Operator ◽  
Complex Banach Space ◽  
Weyl’S Theorem ◽  
Continuous Linear ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Operator Equations ◽  
Weyl Spectrum

A continuous linear operator on a complex Banach space is said to be paranormal if ‖Tx‖2 ≤ ‖T2x‖ ‖x‖ for all x ∈ X. T is called totally paranormal if T–λ is paranormal for every λ ∈ C. In this paper we investigate the class of totally paranormal operators. We shall see that Weyl's theorem holds for operators in this class. We also show that for totally paranormal operators the Weyl spectrum satisfies the spectral mapping theorem. In Section 5 of this paper we investigate the operator equations eT = eS and eTeS = eSeT for totally paranormal operators T and S.

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 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.

Semigroups of operators and an application to spectral theory

1984 ◽  
Vol 96 (1) ◽  
pp. 143-149 ◽  
Author(s):  
W. Ricker
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Spectral Theory ◽  
Continuous Linear Operator ◽  
Integral Transforms ◽  
Fundamental Importance ◽  
Semigroups Of Operators ◽  
Continuous Linear ◽  
Scalar Type

A problem of fundamental importance in Spectral Theory consists of finding criteria for an operator to be of scalar-type in the sense of N. Dunford [1]. One relatively general approach in determining such criteria is based on the method of integral transforms (see for example [4], [5], [6], [11], [12]). For example, if X is a Banach space and T is a continuous linear operator on X, then the group {eitT; t real} exists. As noted by several authors (e.g. [4], [6]), this group can then be effectively used for analysing the operator T.

scholarly journals Construction of operators with prescribed orbits in Fréchet spaces with a continuous norm

2011 ◽  
Vol 109 (1) ◽  
pp. 147 ◽  
Author(s):  
Angela A. Albanese
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Fréchet Space ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Continuous Norm ◽  
Infinite Dimensional ◽  
Linearly Independent ◽  
Dense Set

Let $X$ be a separable, infinite dimensional real or complex Fréchet space admitting a continuous norm. Let $\{v_n:\ n\geq 1\}$ be a dense set of linearly independent vectors of $X$. We show that there exists a continuous linear operator $T$ on $X$ such that the orbit of $v_1$ under $T$ is exactly the set $\{v_n:\ n\geq 1\}$. Thus, we extend a result of Grivaux for Banach spaces to the setting of non-normable Fréchet spaces with a continuous norm. We also provide some consequences of the main result.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

Sign in / Sign up

Export Citation Format

Share Document