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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An automatic adjoint theorem and its applications


Authors: Junde Wu and Shijie Lu
Journal: Proc. Amer. Math. Soc. 130 (2002), 1735-1741
MSC (2000): Primary 46A45, 47A05
Published electronically: December 20, 2001
MathSciNet review: 1887021
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Abstract: In this paper, we prove the following automatic adjoint theorem: For any sequence spaces $E(X)$ and $F(Y)$, if $E(X)$ has the signed-weak gliding hump property and $A$ is an infinite matrix which transforms $E(X)$into $F(Y)$, then the transpose matrix $A'$ of $A$ transforms $F(Y)^\beta$into $E(X)^\beta$, and for any $x\in E(X)$ and $T\in F(Y)^\beta$, $[Ax,T]=[x,A'T]$. That is, the adjoint operator of $A$ automatically exists and is just the transpose matrix $A'$ of $A$. From the theorem we obtain a class of infinite matrix topological algebras $(\lambda,\mu)$, and prove also a $\lambda$-multiplier convergence theorem of Orlicz-Pettis type. The theorem improves substantially the famous Stiles' Orlicz-Pettis theorem.


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Additional Information

Junde Wu
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China
Email: wjd@math.zju.edu.cn

Shijie Lu
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06285-2
PII: S 0002-9939(01)06285-2
Keywords: Sequence space, infinite matrix, adjoint operator
Received by editor(s): June 27, 2000
Received by editor(s) in revised form: December 7, 2000
Published electronically: December 20, 2001
Additional Notes: This research was partially supported by the National Natural Science Foundation of China
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2001 American Mathematical Society