Mean separations in Banach spaces under abstract interpolation and extrapolation

Abstract

logarithmic space a b s t r a c tWe introduce the mean separation for bounded sequences in Banach spaces and the related seminorm for bounded linear operators. The introduced quantities are closely related to the geometric characterizations of the Banach-Saks property and the alternate signs Banach-Saks property. We investigate the behavior of the mean separations for a class of operators between vector-valued Banach sequence spaces E(X ν ), providing that a Banach sequence lattice E has the Banach-Saks property. We estimate the mean separations for operators under abstract interpolation and extrapolation methods. In particular, we obtain quantitative and qualitative results on the heredity of the Banach-Saks properties under these methods.

Quantification of the Banach–Saks property

“…This is inspired by a large number of recent results on quantitative versions of various theorems and properties of Banach spaces, see, e.g., [3,2,10,11,16,18,20,8]. Another approach to quantification of the Banach-Saks property and some related properties was followed by A. Kryczka in a recent series of papers [22,23,24,25]. More precisely, in the quoted papers mainly quantitative versions of Banach-Saks and weak Banach-Saks operators are investigated.…”

Deviation from weak Banach–Saks property for countable direct sums

“…In the main result, we show that the deviation from the WBS property of an operator is equal to the supremum of such deviations attained on the coordinates X ν , providing that a Banach sequence lattice E has the Banach-Saks property. Our main tool in the proofs is a repeated averaging technique elaborated in [7,8], and based on the spreading models of Brunel and Sucheston [3].…”