Measure of weak noncompactness under complex interpolation

Abstract

Abstract. Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón's complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T : [θ] for all 0 < θ < 1, where A [θ] and B [θ] are interpolation spaces with respect to the pairs (A 0 , A 1 ) and (B 0 , B 1 ). Some formulae for this measure and relations to other quantities measuring weak noncompactness are established. are reflexive for all 0 < θ < 1 and 1 < p < ∞. In this paper we consider Calderón's complex interpolation. The counterpart of Beauzamy's result is false for this interpolation method (see [25]). Nevertheless, Calderón [10] proved that if one of the Banach spaces A 0 , A 1 is reflexive then so is the interpolation space A [θ] for every 0 < θ < 1. Introduction. Measures of noncompactness or weak noncompactIn [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). Namely, for all 0 < θ < 1 and 1 < p < ∞ the following estimate was established:

Lions–Schechter's methods of complex interpolation and some quantitative estimates

“…In this section, let us first consider the measure of weak noncompactness for bounded linear operators given by Kryczka and Prus [9], and apply their idea on the C (±n) -methods. Let X be a Banach space and let x 1 be a sequence in X. Vectors u 1 , u 2 in X are said to be a pair of successive convex combinations (scc for short) for x if u 1 ∈ conv x k =1 and u 2 ∈ conv x ∞ =k+1 for some integer k 1.…”

Measures of weak noncompactness and fixed point theory for 1-set weakly contractive operators on unbounded domains

“…27, No.3 (2011) 227 Let us mention that in the paper [4] a measure of weak non compactness in the above sense is called to be regular. For more examples and properties of measures of weak noncompactness we refer to [1,4,5,14,15].…”

“…The notion of the measure of weak noncompactness was introduced by De Blasi in 1977, see [9]). This index has found applications in operator theory (see [14,15]) and many existence results for weak solutions of differential and integral equations in Banach spaces (see [8,20,25] and other). Recall that weak solutions of the Cauchy problem in reflexive Banach spaces were investigated by Szép [25] and weak solutions of nonlinear integral equations in these spaces by O'Regan [20] .…”

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.…”