Measure of weak noncompactness and real interpolation of operators

Abstract

A new measure of weak noncompactness is introduced. A logarithmic convexity-type result on the behaviour of this measure applied to bounded linear operators under real interpolation is proved. In particular, it gives a new proof of the theorem showing that if at least one of the operators T : Ai -» Bi, i = 0,1 is weakly compact, then so is T : A g<p -> B e ,p for all 0 < 6 < 1 and 1 < p < oo.

On an integral equation under Henstock–Kurzweil–Pettis integrability

“…For more examples and properties of measures of weak noncompactness, we refer the reader to [2,4,5,21,22]. Definition 2.2 A function f : X 1 −→ X 2 , where X 1 and X 2 are Banach spaces, is said to be weakly-weakly sequentially continuous if for each weakly convergent (x n ) n ⊂ X 1 with x n x, we have f x n f x.…”

Measure of weak noncompactness under complex interpolation

“…From our point of view, the crucial fact about sequences of scc is the following theorem based on an idea of Milman [26]. For the convenience of the reader we repeat the proofs of the next two theorems from [24].…”

“…The notion of scc was used in [24] to define a measure of weak noncompactness γ which is a counterpart for the weak topology of the separation measure of noncompactness (see [1], [5]). By the convex separation of (x n ) we mean csep(x n ) = inf{ u 1 − u 2 : u 1 , u 2 is a pair of scc for (x n )}.…”

“…Condition (1) is a consequence of James' criterion of weak compactness (see [22]). Using an idea from [22] one can also obtain the following formulae [24].…”

“…In [24], a measure of weak noncompactness γ for sets and a corresponding measure Γ for operators were introduced. The measure Γ was applied to the Lions-Peetre real interpolation method (in a discrete form).…”