scholarly journals Remark on our paper ``On bases in Banach spaces'' (Studia Math. 170 (2005), 147–171)

2009 ◽  
Vol 191 (2) ◽  
pp. 201
Author(s):  
T. Bartoszyński ◽  
M. Džamonja ◽  
L. Halbeisen ◽  
E. Murtinová ◽  
A. Plichko
Keyword(s):  
Banach Spaces ◽  
Studia Math

scholarly journals ON λ-STRICT IDEALS IN BANACH SPACES

2010 ◽  
Vol 83 (2) ◽  
pp. 231-240 ◽  
Author(s):  
TROND A. ABRAHAMSEN ◽  
OLAV NYGAARD
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Studia Math ◽  
Separable Case ◽  
Baire One ◽  
Ideal Property ◽  
Open Question

AbstractWe define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

scholarly journals Invariant vector means and complementability of Banach spaces in their second duals

2022 ◽  
Author(s):  
Radosław Łukasik
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Studia Math ◽  
Cancellative Semigroup ◽  
Invariant Mean ◽  
Invariant Vector ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractLet X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of $$X^{**}$$ X ∗ ∗ . We then show that X is complemented in $$X^{**}$$ X ∗ ∗ if and only if there exists an invariant mean $$M:\ell _\infty (S,X)\rightarrow X$$ M : ℓ ∞ ( S , X ) → X . This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).

scholarly journals Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings

2020 ◽  
Vol 53 (1) ◽  
pp. 38-43 ◽  
Author(s):  
Moosa Gabeleh ◽  
Hans-Peter A. Künzi
Keyword(s):  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Studia Math ◽  
Normal Structure ◽  
Projection Algorithm ◽  
Best Proximity Points ◽  
Iteration Sequence ◽  
Cyclic Contractions ◽  
Convergence Method ◽  
Strictly Convex Banach Spaces

AbstractIn this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

ON HYPERSTABILITY OF ADDITIVE MAPPINGS ONTO BANACH SPACES

2014 ◽  
Vol 91 (2) ◽  
pp. 278-285 ◽  
Author(s):  
YUNBAI DONG ◽  
BENTUO ZHENG
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Studia Math ◽  
Additive Map ◽  
Image Position ◽  
Additive Mappings

AbstractLet$(X,+)$be an Abelian group and$E$be a Banach space. Suppose that$f:X\rightarrow E$is a surjective map satisfying the inequality$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(x-y)\Vert \,|\leq {\it\varepsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(x-y)\Vert ^{p}\}\end{eqnarray}$$for some${\it\varepsilon}>0$,$p>1$and for all$x,y\in X$. We prove that$f$is an additive map. However, this result does not hold for$0<p\leq 1$. As an application, we show that if$f$is a surjective map from a Banach space$E$onto a Banach space$F$so that for some${\it\epsilon}>0$and$p>1$$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(u)-f(v)\Vert \,|\leq {\it\epsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(u)-f(v)\Vert ^{p}\}\end{eqnarray}$$whenever$\Vert x-y\Vert =\Vert u-v\Vert$, then$f$preserves equality of distance. Moreover, if$\dim E\geq 2$, there exists a constant$K\neq 0$such that$Kf$is an affine isometry. This improves a result of Vogt [‘Maps which preserve equality of distance’,Studia Math.45(1973) 43–48].

scholarly journals A short proof that ℬ(L1) is not amenable

2020 ◽  
pp. 1-10
Author(s):  
Yemon Choi
Keyword(s):  
Banach Spaces ◽  
Short Proof ◽  
Studia Math ◽  
Transference Principle ◽  
The Ideal

Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓ p and E = L p for all 1 ⩽ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of $\ell ^\infty ({\mathcal {K}}(\ell _1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010). In this note, we provide a short proof that ${\mathcal {B}}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that ${\mathcal {B}}(L_1)$ is not even approximately amenable.

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