18.4 Adjoint Maps

Definition 18.4.1 (Adjoint Map).label Let $E, F$ be vector spaces over a field $K$, and $T \in \hom(E; F)$ be a linear map, then the mapping

is the algebraic adjoint of $T$.

Proposition 18.4.2.label Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$ and $T \in \hom(E; G)$, then the following are equivalent:

1. (1)

   $T$ is $\sigma(E, F)$-$\sigma(G, H)$ continuous.

2. (2)

   $T^{*}(H) \subset F$.

If the above holds, then

1. (3)

   $T^{*}|_{H}$ is $\sigma(H, G)$-$\sigma(F, E)$ continuous.

2. (4)

   $T^{**}= T$.

and the restriction of $T^{*}$ to $H$ is the adjoint of $T$ with respect to $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$.

Proposition 18.4.3.label Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$, $T: E \to G$ be a $\sigma(E, F)$-$\sigma(G, H)$ continuous linear map, $A \subset E$, and $B \subset G$, then:

1. (1)

   $T(A)^{\circ} = (T^{*})^{-1}(A^{\circ})$.

2. (2)

   If $T(A) \subset B$, then $T^{*}(B^{\circ}) \subset A^{\circ}$.

Corollary 18.4.4.label Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$ and $T: E \to G$ be a $\sigma(E, F)$-$\sigma(G, H)$ continuous linear map, then:

1. (1)

   $\ker(T^{*}) = T(E)^{\perp} = \bracs{\phi \in H| \dpn{y, \phi}{\mu} = 0 \forall y \in T(E)}$.

2. (2)

   $T^{*}$ is injective if and only if $T(E)$ is $\sigma(G, H)$-dense in $G$.

Proposition 18.4.5.label Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$, $T: E \to G$ be a $\sigma(E, F)$-$\sigma(G, H)$ continuous linear map, $\sigma \subset 2^{E}$ be a saturated ideal of $\sigma(E, F)$-bounded sets, $\tau \subset 2^{G}$ be a saturated ideal of $\sigma(G, H)$-bounded sets, then the following are equivalent:

1. (1)

   $T^{*}$ is continuous with respect to the $\tau$-uniform topology on $H$ and the $\sigma$-uniform topology on $F$.

2. (2)

   $T(\sigma) \subset \tau$.