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Representation theorems for compact operators

Author: Daniel J. Randtke
Journal: Proc. Amer. Math. Soc. 37 (1973), 481-485
MSC: Primary 47B05; Secondary 46B15
MathSciNet review: 0310685
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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}$.

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Keywords: Normed space, Banach space, Fréchet space, compact linear operator, precompact linear operator, unconditionally summable sequence, weakly unconditionally summable sequence, equicontinuous sequence, normalized basis, associated sequence of coefficient forms
Article copyright: © Copyright 1973 American Mathematical Society

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