On Operators Acting on Convergent Sequences in Banach Spaces

✍ Scribed by K. Peter Cass

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 335 KB
• Volume: 180
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.3211800102

✦ Synopsis

Our concern is to find a representation theorem for operators in B ( c ( X ) , c ( Y ) ) where S and Y are Banach spaces with Y containing an isomorphic copy of Q. CASS and GAO [l] obtained a iq,resentation theorem that always applies if Y does not contain an isomorphic copy of Q. MADDOX [:$I, MELVIN-MELVIN [4], and ROBINSON [5] consider Operators in B ( c ( X ) , c ( Y ) ) that are given by matrices. In this paper we show that CASS'S and GAO'S result in [I] can be extended, when Y contains an isomorphic copy of Q, to certain operators that we call representable. In addition, we ~l i o w that when Y contains an isomorphic copy of Q there are always operators that fall outside the Hcope of our representation theorem. Light is also cast on a theorem given in MADDOX [3, Theorem 4.21 and (5, Theorem IV]. * A ' and Y be Banach spaces and A = {Ank} be an infinite matrix of linear, but not 1991 Mathematics Subject Classification. Primary 47B37; Secondary 40H05. Keywonia and phmses. Banach Sequence space, Representation of Operators.

'llhh, operators in Banach Sequence Spaces CASS. P., and GAO, X.: Bounded Linear Operators on Convergent Sequences with Terms in a B a n d Space, Analysis 13 (1993), 85 -101