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.
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.
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.
AbstractIn this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space $\ell _{1}$ℓ1 of all absolutely summable sequences. Examples of nuclear operators over the space $\ell _{1}$ℓ1 are given and used to construct operators over general Banach spaces with specific approximation numbers.
AbstractIf A is a norm closed algebra of compact operators on a Hilbert space and if its Jacobson radical J(A) consists of all quasinilpotent operators in A then A/ J(A) is commutative. The result is not valid for a general algebra of polynomially compact operators.
On strong convex compactness property of spaces of nonlinear operators
