Pap adjoint theorem

If $(X,\tau)$ is a topological vector space, a sequence $\{x_k\}$ in X is said to be a $\tau$-$K$ sequence if every subsequence of $\{x_k\}$ has a further subsequence $\{x_{n_k}\}$ such that the subseries $\sum_kx_{n_k}$ is $\tau$-convergent to an element of $X$.

A topological vector space $(X,\tau)$ is said to be a $K$-space if every sequence which converges to $0$ is a $\tau$-$K$ sequence. A subset $A$ of $X$ is said to be $\tau$-$K$ bounded if for every sequence $\{x_n\}\subseteq A$ and every scalar sequence $\{t_n\}$ converging to $0$, the sequence $\{t_nx_n\}$ is a $\tau$-$K$ sequence.

Let $E$ and $F$ be Hausdorff locally convex topological vector spaces (cf. also Locally convex space; Hausdorff space) and let $T:E\to F$ be a linear mapping. The domain of the adjoint operator, $T'$, is defined to be

\begin{equation}D(T')=\{y'\in F':y'T\in E'\}\end{equation}

and $T':D(T')\to E'$ is defined by $T'y'=y'T$.

The first adjoint theorem was proved by E. Pap [a3] for operators defined on pre-Hilbert $K$-spaces. There exists a pre-Hilbert $K$-space which is not a Hilbert space, [a2]. A generalization of the adjoint theorem for normed spaces was given in [a1], [a4]. It reads as follows.

Let $E$ be a normed $K$-space, let $F$ be a normed space and let $T:E\to F$ be a linear operator. Then the adjoint operator $T'$ is a bounded linear operator on $D(T')$.

In the proofs of all these theorems, so-called diagonal theorems were used (cf. also Diagonal theorem). As a simple consequence, a proof of the closed-graph theorem without the Baire category argumentation was obtained, [a3], [a4], [a6], [a8], [a9].

There is a locally convex generalization of the adjoint theorem [a5], [a7]: $T'$ is sequentially continuous with respect to the relative $\sigma(F',F)$- (weak) topology on $D(T')$ and the topology on $E'$ of uniform convergence on $\sigma(E,E')$-$K$-convergent sequences. In particular, $T'$ is bounded with respect to these topologies.

A special case is obtained when $E$ is a normed $K$-space. Then $T'$ maps weak-$*$ bounded subsets of $D(T')$ to norm-bounded subsets of $E'$. In particular, $T'$ is norm-bounded.

References

[a1]	P. Antosik, Swartz, C., "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)
[a2]	C. Kliś, "An example of non-complete normed (K)-space" Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys. , 26 (1976) pp. 415–420
[a3]	E. Pap, "Functional analysis with K-convergence" , Proc. Conf. Convergence, Bechyne, Czech. , Akad. Berlin (1984) pp. 245–250
[a4]	E. Pap, "The adjoint operator and K-convergence" Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. , 15 : 2 (1985) pp. 51–56
[a5]	E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995)
[a6]	E. Pap, C. Swartz, "The closed graph theorem for locally convex spaces" Boll. Un. Mat. Ital. , 7 : 4-B (1990) pp. 109–111
[a7]	E. Pap, C. Swartz, "A locally convex version of adjoint theorem" Univ. u Novom Sadu Zb. Rad. Prirod. - Mat. Fak. Ser. Mat. , 24 : 2 (1994) pp. 63–68
[a8]	C. Swartz, "The closed graph theorem without category" Bull. Austral. Math. Soc. , 36 (1987) pp. 283–288
[a9]	C. Swartz, "Introduction to functional analysis" , M. Dekker (1992)