Pei-Kee Lin

Köthe-Bochner Function Spaces

In this chapter, we introduce the Kothe-Boehner function spaces, and provide some basic results in this area.

Remarks about Schlumprecht space

Let S denote the Schlumprecht space. We prove that 1. l∞ is finitely disjointly respresentable in S; 2. S contains an l1-spreading model; 3. for any sequence (nk) of natural numbers, S is isomorphic to the space ( ∑∞ k=1⊕ lnk ∞ ) S . Let (ei) ∞ i=1 be the standard basis of the linear space c00, the set of all finitely supported sequences. For x = ∞ ∑ i=1 aiei ∈ c00, suppx denotes the set {i ∈ N : ai 6= 0}. A subset E of N is said to be an interval if there exist a, b such that E = {c ∈ N : a < c < b}. For finite subsets E,F of N, E < F means maxE < minF or E is an empty set. For x = ∞ ∑ i=1 aiei and a subset E of N, Ex denotes the vector Ex = ∑ i∈E aiei. Let f : [1,∞) → [1,∞) be the function defined by f(x) = log2(x+1). The Schlumprecht space S = (S, ‖ · ‖) is the completion of c00 with respect to the norm ‖ · ‖ which satisfies the following implicit equation: (1) ‖x‖ = max { ‖x‖∞, sup E1

Unrestricted products of contractions in Banach spaces

Let

Property

X

(H)

be a reflexive Banach space such that for any

in Lebesgue-Bochner function spaces

x \ne 0

Stability of the fixed point property of Hilbert spaces

the set

Some remarks of drop property

\{x^* \in X^*: \text {

Norm closed ideals in the algebra of bounded linear operators on Orlicz sequence spaces

\|x^*\|=1