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AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES

Published online by Cambridge University Press:  13 August 2013

ANIL KUMAR KARN  and
DEBA PRASAD SINHA
ANIL KUMAR KARN
Affiliation:
School of Mathematical Sciences, National Institute of Science Education and Research, Institute of Physics Campus, P.O. Sainik School, Bhubaneswar 751005, Odisha, India e-mail: anilkarn@niser.ac.in
DEBA PRASAD SINHA
Affiliation:
Department of Mathematics, Dyal Singh College (University of Delhi), Lodi Road, New Delhi 110003, India e-mail: sinha.deba@gmail.com
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Abstract

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Let 1 ≤ p < ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.

Keywords

Primary 46B20Secondary 46B2846B50
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 427 - 437
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

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