Adjoint operator

Adjoint operator on topological vector spaces

Adjoint operator can be easily defined over the general vector spaces. However, defining it with respect to topological vector spaces (TVS) is non-trivial. This post clarifies the conditions for such an adjoint:

• The topological vector spaces should be locally convex Hausdorff
• The original operator is weakly continuous

The main reference is [1].

Adjoint operator for general vector spaces

Let $X$ be a general vector space and $X'$ be its algebraic dual space, which contains all linear functionals on $X$. Similarly for $Y$ and $Y'$. Let $A: X \longrightarrow Y$. The algebraic adjoint $A': Y' \longrightarrow X'$ is a linear operator defined as follows, for all $x\in X$ and $y'\in Y'$,
$$A'y': x \in X \longmapsto \langle y', Ax\rangle\in \mathbb{R}$$
It can also be rewritten
$$\langle y', Ax\rangle_{Y', Y} = \langle A'y', x\rangle_{X', X} \tag{adjoint}$$
We may drop the space subscripts since it is clear from the context.

Adjoint operator for topological vector spaces

Let $(X, \tau_X)$ be a locally convex Hausdorff space and $X^*$ be its continuous dual, i.e. the space of continuous linear functionals defined on $X$. By Example 8.1.1. d) [1], we know that $(X, X^*)$ is a dual pair, i.e. $X$ distinguishes points of $X^*$ and vice versa. Note that general topological vector space $(X, \tau_X)$ may not ensure the paired duality of $X$ and $X^*$.

Let $\omega_X=\sigma(X, X^*)$ be the coarse topology so that each element in $X^*$ can be considered as continuous over $(X, \sigma(X, X^*))$, this topology is also known as the weak topology on $X$. Similarly the topology $\omega_{X^*}=\sigma(X^*, X)$ on $X^*$ is called weak-$*$ topology.

Similar notations can be applied for $Y$ and $Y^*$.

An operator $A: X\longrightarrow Y$ is said to be weak continuous if $A: (X, \omega_X) \longrightarrow (Y, w_Y)$ is continuous. An operator $B: Y^*\longrightarrow X^*$ is said to be weak-$*$ continuous if $B: (Y^*, \omega_{Y^*}) \longrightarrow (X^*, w_{X^*})$ is continuous.

It is easy to see $X^*$ can be embedded into $X'$. So it is natural to define $A^*$ as a restriction of $A'$ onto $X^*$. More precisely, $A^*: Y^*\longrightarrow X'$ so that $$A^* = A'|_{Y^*}$$
However, to define $A^*: Y^*\longrightarrow X^*$, it necessary that the range of $A^*$ should be a subset of $X^*$. This is not true for a general operator $A$. The following theorem tells us that it only happens if $A$ is weakly continuous.

Theorem 1. (Theorem 8.10.3 [1]) Let $X, Y$ be locally convex Hausdorff spaces. Let $X^*$ and $Y^*$ be, respectively, the continuous dual space of $X$ and $Y$. Let $A^* = A'|_{Y^*}$, then $A^*(Y^*)\subset X^*$ iff $A$ is weakly continuous linear operator.

By this theorem, we may define $A^*: Y^*\longrightarrow X^*$ satisfying $(\text{adjoint})$ as soon as $A$ is weakly continuous.

Remark. Every normed space is locally convex since the unit ball is convex. Moreover, the normed spaces are Hausdorff since every metric space is Hausdorff.

Continuity of adjoint operator

Assume that $A$ is weakly continuous, so $A^*$ is well-defined.

Theorem 2. (Theorem 8.10.5 [1]) $A^*$ is weakly-$*$ continuous and $A^{**}=A$

Theorem 3. (Theorem 6 page 165 [3]) Let $X, Y$ be locally convex separated spaces and $K: Y^* \longrightarrow X^*$ a linear map. Then there exists $A: X\longrightarrow Y$ with $K=A^*$ if and only if $K$ is weak-$*$ continuous.

Theorem 4. (Corollary 8.11.4 [1]) If $X$ and $Y$ are normed spaces, $A$ is continuous iff it is weakly continuous.

Theorem 5. (Theorem 8.11.5 [1]) Let $X, Y$ be normed spaces. If $A$ is continuous then $A^*$ is also continuous and $||A||=||A^*||$.

Theorem 6. (Schauder’s Theorem see Theorem 2 in [4] epage 499 page 485, Theorem 4.19 [2]) Let $X, Y$ be Banach spaces and $A$ is continuous. $A$ is compact iff $A^*$ is compact.

References

[1]: Narici, Lawrence, and Edward Beckenstein. Topological vector spaces. Chapman and Hall/CRC, 2010.
[2]: Rudin, Walter. Functional Analysis
[3]: Wilansky, Albert. Modern methods in topological vector spaces. 1978
[4]: Dunford, Nelson, and Jacob T. Schwartz. Linear operators, part 1: general theory. Vol. 10. John Wiley & Sons, 1988.