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.

Abstract. We prove that there is c > 1 such that the unit ball of any nonreflexive Banach space contains a c-separated sequence. The supremum of these constants c is estimated from below by 5 √ 4 and from above approximately by 1.71. Given any p > 1, we also construct a nonreflexive space so that if the convex hull of a sequence is sufficiently close to the unit sphere, then its separation constant does not exceed 2

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We study a quantitative approach to weak noncompactness of operators under the Cobos-Peetre polygon interpolation method for Banach TV-tuples. In the case of operators acting between two J-spaces or two if-spaces obtained by this method we prove logarithmically convex-type inequalities for certain operator seminorm vanishing on the subspace of weakly compact operators. Geometrically speaking, in these estimates only some triangles inscribed in the polygon are involved. For operators acting from a J-space to a K-space we prove logarithmically convex-type estimates where all polygon vertices are included. In particular, the estimates obtained here give the new proofs of the results showing the relation between distribution of weakly compact operators among polygon vertices and weak compactness of operators under interpolation.