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

B e ,p for all 0 < 6 < 1 and 1 < p < oo.

Abstract. In the space of bounded linear operators acting between Banach spaces we define a seminorm vanishing on the subspace of operators having the alternate signs Banach-Saks property. We obtain logarithmically convex-type estimates of the seminorm for operators interpolated by the Lions-Peetre real method. In particular, the estimates show that the alternate signs BanachSaks property is inherited from a space of an interpolation pair (A 0 , A 1 ) to the real interpolation spaces A θ,p with respect to (A 0 , A 1 ) for all 0 < θ < 1 and 1 < p < ∞.

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:

We introduce the arithmetic separation of a sequence-a geometric characteristic for bounded sequences in a Banach space which describes the Banach-Saks property. We define an operator seminorm vanishing for operators with the Banach-Saks property. We prove quantitative stability of the seminorm for a class of operators acting between lp-sums of Banach spaces. We show logarithmically convex-type estimates of the seminorm for operators interpolated by the real method of Lions and Peetre. Mathematics Subject Classification (2000). Primary 46B70, 47A30; Secondary 47B10.