scholarly journals Representing completely continuous operators through weakly ∞-compact operators

2016 ◽  
Vol 48 (3) ◽  
pp. 452-456 ◽  
Author(s):  
William B. Johnson ◽  
Rauni Lillemets ◽  
Eve Oja
Keyword(s):  
Compact Operators ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Continuous Operators ◽  
Weakly Compact Operators ◽  
Continuous Operators

Continuous linear operators on Orlicz-Bochner spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1147-1155 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Finite Measure ◽  
Compact Operators ◽  
Linear Operators ◽  
Continuous Linear ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Continuous Operators ◽  
Weakly Compact Operators ◽  
Continuous Operators ◽  
Continuous Linear Operators

Abstract Let (Ω, Σ, μ) be a complete σ-finite measure space, φ a Young function and X and Y be Banach spaces. Let Lφ(X) denote the corresponding Orlicz-Bochner space and $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$ denote the finest Lebesgue topology on Lφ(X). We examine different classes of ( $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$, ∥ ⋅ ∥Y)-continuous linear operators T : Lφ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The relationships among these classes of operators are established.

The Uncomplemented Subspace K(X,Y)

2013 ◽  
Vol 56 (1) ◽  
pp. 65-69
Author(s):  
Ioana Ghenciu
Keyword(s):  
Vector Measure ◽  
Compact Operators ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Spaces Of Operators ◽  
Completely Continuous Operators ◽  
Weakly Compact Operators ◽  
Continuous Operators

AbstractA vector measure result is used to study the complementation of the space K(X,Y) of compact operators in the spaces W(X,Y) of weakly compact operators, CC(X,Y) of completely continuous operators, and U(X,Y) of unconditionally converging operators. Results of Kalton and Emmanuele concerning the complementation of K(X,Y) in L(X,Y) and in W(X,Y) are generalized. The containment of c0 and ℓ∞ in spaces of operators is also studied.

scholarly journals On strong convex compactness property of spaces of nonlinear operators

2006 ◽  
Vol 74 (3) ◽  
pp. 411-418
Author(s):  
Xueli Song ◽  
Jigen Peng
Keyword(s):  
Compact Operators ◽  
Compactness Property ◽  
Operator Semigroups ◽  
Weakly Compact ◽  
Nonlinear Operators ◽  
Completely Continuous ◽  
Completely Continuous Operators ◽  
Weakly Compact Operators ◽  
Convex Compactness ◽  
Continuous Operators

The strong convex compactness property is important for property persistence of operator semigroups under perturbations. It has been investigated in the linear setting. In this paper, we are concerned with the property in the nonlinear setting. We prove that the following spaces of (nonlinear) operators enjoy the strong convex compactness property: the space of compact operators, the space of completely continuous operators, the space of weakly compact operators, the space of conditionally weakly compact operators, the space of weakly completely continuous operators, the space of demicontinuous operators, the space of weakly continuous operators and the space of strongly continuous operators. Moreover, we prove the property persistence of operator semigroups under nonlinear perturbation.

Complemented Subspaces of Linear Bounded Operators

2012 ◽  
Vol 55 (3) ◽  
pp. 449-461 ◽  
Author(s):  
Manijeh Bahreini ◽  
Elizabeth Bator ◽  
Ioana Ghenciu
Keyword(s):  
Compact Operators ◽  
Linear Operators ◽  
Bounded Operators ◽  
Weakly Compact ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Complemented Subspaces ◽  
Completely Continuous Operators ◽  
Grothendieck Space ◽  
Continuous Operators

AbstractWe study the complementation of the space W(X, Y) of weakly compact operators, the space K(X, Y) of compact operators, the space U(X, Y) of unconditionally converging operators, and the space CC(X, Y) of completely continuous operators in the space L(X, Y) of bounded linear operators from X to Y. Feder proved that if X is infinite-dimensional and c0 ↪ Y, then K(X, Y) is uncomplemented in L(X, Y). Emmanuele and John showed that if c0 ↪ K(X, Y), then K(X, Y) is uncomplemented in L(X, Y). Bator and Lewis showed that if X is not a Grothendieck space and c0 ↪ Y, then W(X, Y) is uncomplemented in L(X, Y). In this paper, classical results of Kalton and separably determined operator ideals with property (∗) are used to obtain complementation results that yield these theorems as corollaries.

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