An automatic adjoint theorem and its applications

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.