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 is an element of E(X) and T is an element of 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.