Uniform $L\sb p(w)$ spaces

Abstract

$L_p(w)$ spaces ($0<p<1$) were developed by J. W. Roberts to serve as a special class of trivial-dual spaces which admit compact operators and to provide counterexamples to various interesting problems. Roberts showed that any separable, trivial-dual $p$-Banach space is a quotient of some \emph{uniform $L_p(w)$ space}. Uniform $L_p(w)$ spaces are indexed by a sequence of finite dimensional spaces $\langle X_n \rangle$ in $L_p$ and a sequence of constants $\langle c_n \rangle$ such that $1 \leq c_0 \leq c_1 \leq c_2 \leq \cdots$. If $\langle c_n \rangle$ is bounded, the resulting space is isomorphic to $L_p$. Hence these spaces can be thought of as generalized $L_p$ spaces. We prove that if $c_n \uparrow \infty$, the corresponding $L_p(w)$ space admits compact operators and is thus not isomorphic to $L_p$. Further, we show that there is no non-zero continuous linear operator from $L_p$ into any $L_p(w)$, where $ c_n \uparrow \infty$. Using and sharpening a result of Roberts, we also demonstrate that for any separable, trivial-dual $p$-Banach space $S$ there exists a uniform $L_p(w)$ space $X_S$ with $\mathcal{L} (S,X_S)=\{0\}$.