Alternate signs Banach-Saks property and real interpolation of operators

Abstract

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 < ∞.

Deviation from weak Banach–Saks property for countable direct sums

“…The next result is a part of Proposition 2.3 of [7], where the proof based on spreading models was given for a similar characteristics of a sequence related to the alternate signs Banach-Saks property. The proof for ψ runs in much the same way.…”

“…[12,Theorem 2], Ψ(T ) = 0 if and only if T has the WBS property. Applying Proposition 2, we can show, as in the proof of Proposition 2.5 of [7], that Ψ is a seminorm in L(X, Y ). The procedure of stabilization of ψ plays a key role also in the next result.…”

“…The procedure of stabilization of ψ plays a key role also in the next result. The arguments of the proof are similar to those used in the proofs of Theorem 3 of [12] and Theorem 3.2 of [7]. …”

“…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].…”

Mean separations in Banach spaces under abstract interpolation and extrapolation

“…The notion we introduce is flexible and can be used as a separation of a mean from zero or between successive means. In particular, the mean separations applied to operators give quantities which include the measures of deviation from the ABS and BS properties considered in [24,25].…”