Representation theorems for compact operators

Abstract

It is shown that c0 (the Banach space of zeroconvergent sequences) is the only Banach space with basis that satisfies the following property: For every compact operator T:c"-->-E from c0 into a Banach space E, there is a sequence X in c0 and an unconditionally summable sequence {yn} in E such that Tn=J An/i"j" for each /i in c0. This result is then used to show that a linear operator T:E-*F from a locally convex space E into a Fréchet space Fhas a representation of the form Tx= y A"<x, a")j>", where A is a sequence in c", {o"} is an equicontinuous sequence in the topological dual E' of E and {y,} is an unconditionally summable sequence in F, if and only if Tcan be "compactly factored" through c0.

On a theorem of Terzioğlu

“…Another related representation theorem for compact maps emphasizing the factorization through c 0 can be found in [17]. Using all of the above results of Terzioğlu, the following conclusions are shown in [20].…”