Fixed point property of direct sums

WORTH property, García-Falset coefficient and Opial property of infinite sums

“…Also, by [11,Corollary 3.17] and the remarks after [11, Definition 1.5] X 2 is again a U -space, so R(X 2 ) < 2. From the aforementioned result [4,Theorem 7] it follows that R(X 4 ) < 2 and since Theorem 7] implies that R(X) < 2.…”

“…Unfortunately, the author does not know whether the simpler condition β := sup i∈I R(X i ) < 2 would be already enough to ensure that α < 2. The proof of [4,Theorem 7] shows that for β < 2 one has for every finite subset J ⊆ I with |J| = N that R( i∈J X i E ) ≤ 2 − δ, where first ε > 0 is chosen such that β(1 + N ε) < 2 and then 0 < δ < min{2δ E (ε), 2 − β(1 + N ε)}, so it still might be that R( i∈J X i E ) tends to 2 for N → ∞. Next we will discuss some applications of Theorem 3.2.…”

CHARACTERIZATIONS OF GEOMETRICAL PROPERTIES OF BANACH SPACES USING Ψ-Direct SUMS

“…Since the introduction of ψ-direct sums of Banach spaces, it has attracted plenty of attention and been treated by several authors (cf. [3,4,5,12,16]). …”

Uniform nonsquareness of direct sums of Banach spaces

“…In their paper they asked if it is possible to generalize their result to direct sums of more than two spaces. A sufficient condition for uniform nonsquareness of direct sums of finite families of Banach spaces can be found in [3]. In this paper we give an answer to the problem posed in [9].…”