Totally smooth renormings
We study the problem of totally smooth renormings of Banach spaces and provide such renormings for spaces which are weakly compactly generated. We also consider renormings for $(a,B,c)$-ideals.
math.FA↗
arXiv subjects
Eve Oja
Publications and source records attributed to Eve Oja.
The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited
Let $X$ and $Y$ be Banach spaces and let $Ω$ be a compact Hausdorff space. Denote by $\mathcal{C}_{p}(Ω,X)$ the space of $p$-continous $X$-valued functions, $1\leq p\leq \infty$. For operators $S\in\mathcal{L}(\mathcal{C}(Ω),\mathcal{L}(X,Y))$ and $U\in\mathcal{L}(\mathcal{C}_{p}(Ω,X),Y)$, we establish integral representation theorems with respect to a vector measure $m:Σ\rightarrow \mathcal{L}(X,Y^{**})$, where $Σ$ denotes the $σ$-algebra of Borel subsets of $Ω$. The first theorem extends the classical Bartle-Dunford-Schwartz representation theorem. It is used to prove the second theorem, which extends the classical Dinculeanu-Singer representation theorem, also providing to it an alternative simpler proof. For the latter (and the main) result, we build the needed integration theory, relying on a new concept of the $q$-semivariation, $1\leq q\leq \infty$, of a vector measure $m:Σ\rightarrow \mathcal{L}(X,Y^{**})$.
math.FA↗
Operators on the Banach space of $p$-continuous vector-valued functions
Let $X$, $Y$, and $Z$ be Banach spaces, and let $α$ be a tensor norm. Let a bounded linear operator $S\in\mathcal{L}(Z,\mathcal{L}(X,Y))$ be given. We obtain (necessary and/or sufficient) conditions for the existence of an operator $U\in\mathcal{L}(Z\hat{\otimes}_αX,Y)$ such that $(Sz)x = U(z\otimes x)$, for all $z\in Z$ and $x\in X$, i.e., $S= U^{#}$, the associated operator to $U$. Let $Ω$ be a compact Hausdorff space and denote by $\mathcal{C}(Ω)$ the space of continuous functions from $Ω$ into $\mathbb{K}$. We apply these results to $S\in\mathcal{L}(\mathcal{C}(Ω),\mathcal{L}(X, Y))$ for characterizing the existence of an operator $U\in\mathcal{L}(\mathcal{C}_{p}(Ω,X),Y)$ such that $U^{#}=S$, where $\mathcal{C}_{p}(Ω,X)$ is the space of $p$-continuous $X$-valued functions, $1\leq p \leq \infty$.
math.FA↗
Weaker relatives of the bounded approximation property for a Banach operator ideal
Fixed a Banach operator ideal $\mathcal A$, we introduce and investigate two new approximation properties, which are strictly weaker than the bounded approximation property (BAP) for $\mathcal A$ of Lima, Lima and Oja (2010). We call them the weak BAP for $\mathcal A$ and the local BAP for $\mathcal A$, showing that the latter is in turn strictly weaker than the former. Under this framework, we address the question of approximation properties passing from dual spaces to underlying spaces. We relate the weak and local BAPs for $\mathcal A$ with approximation properties given by tensor norms and show that the Saphar BAP of order $p$ is the weak BAP for the ideal of absolutely $p^*$-summing operators, $1\leq p\leq\infty$, $1/p + 1/{p^*}=1$.
math.FA↗
On $(p,r)$-null sequences and their relatives
Let $1\leq p < \infty$ and $1\leq r \leq p^\ast$, where $p^\ast$ is the conjugate index of $p$. We prove an omnibus theorem, which provides numerous equivalences for a sequence $(x_n)$ in a Banach space $X$ to be a $(p,r)$-null sequence. One of them is that $(x_n)$ is $(p,r)$-null if and only if $(x_n)$ is null and relatively $(p,r)$-compact. This equivalence is known in the "limit" case when $r=p^\ast$, the case of the $p$-null sequence and $p$-compactness. Our approach is more direct and easier than those applied for the proof of the latter result. We apply it also to characterize the unconditional and weak versions of $(p,r)$-null sequences.
math.FA↗
Principle of local reflexivity respecting subspaces
We obtain a strengthening of the principle of local reflexivity in a general form. The added strength makes local reflexivity operators respect given subspaces. Applications are given to bounded approximation properties of pairs, consisting of a Banach space and its subspace.
math.FA↗