Representation theorems for compact operators

Abstract:

It is shown that ${c_0}$ (the Banach space of zero-convergent sequences) is the only Banach space with basis that satisfies the following property: For every compact operator $T:{c_0} \to E$ from ${c_0}$ into a Banach space E, there is a sequence $\lambda$ in ${c_0}$ and an unconditionally summable sequence $\{ {y_n}\}$ in E such that $T\mu = \sum {\lambda _n}{\mu _n}{y_n}$ for each $\mu$ in ${c_0}$. This result is then used to show that a linear operator $T:E \to F$ from a locally convex space E into a Fréchet space F has a representation of the form $Tx = \sum {\lambda _n}\langle x,{a_n}\rangle {y_n}$, where $\lambda$ is a sequence in ${c_0},\{ {a_n}\}$ is an equicontinuous sequence in the topological dual $E’$ of E and $\{ {y_n}\}$ is an unconditionally summable sequence in F, if and only if T can be “compactly factored” through ${c_0}$.