Advanced Search

We study strict u-ideals in Banach spaces. A Banach space X is a strict u-ideal in its bidual when the canonical decomposition $X***=X*\oplus X^{\perp }$ is unconditional. We characterize Banach spaces which are strict u-ideals in their bidual and show that if X is a strict u-ideal in a Banach space Y then X contains c₀. We also show that ${\ell }_{\infty }$ is not a u-ideal.

We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $\left(S_{\gamma }\right)$ of compact operators on X such that $su{p}_{\gamma }\left|\right|S_{\gamma }T\left|\right|\le \left|\right|T\left|\right|$ and $S_{\gamma }\to I_X$ in the strong operator topology. Similar results for dual spaces are also proved.

Let $X$ be a Banach space. We give characterizations of when $ℱ(Y,X)$ is a $u$-ideal in $𝒲(Y,X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when $ℱ(X,Y)$ is a $u$-ideal in $𝒲(X,Y)$ for every Banach space $Y$, when $ℱ(Y,X)$ is a $u$-ideal in $𝒲(Y,X^{**})$ for every Banach space $Y$, and when $ℱ(Y,X)$ is a $u$-ideal in $𝒦(Y,X^{**})$ for every Banach space $Y$.