Properties of the Adjoint of a Bounded Linear Operator

Criterion for the Adjoint of a Bounded Linear Operator to be Injective

Recall from The Adjoint of a Bounded Linear Operator Between Banach Spaces page that if $X$ and $Y$ are Banach spaces and if $T : X \to Y$ is a bounded linear operator then the adjoint of $T$ is the linear operator $T^* : Y^* \to X^*$ defined for all $f \in Y^*$ by:

(1)
\begin{align} \quad T^*(f) = f \circ T \end{align}

We noted that $T^*$ is also a bounded linear operator from $Y^*$ to $X^*$ and that $\| T^* \| = \| T \|$.

We will now look at some basic properties of adjoints.

 Proposition 1: Let $X$ and $Y$ be Banach spaces and let $S, T : X \to Y$ be bounded linear operators. Then: a) $(T + S)^* = T^* + S^*$. b) $(kT)^* = kT^*$ for all $k \in \mathbb{R}$.
• Proof of a) Let $f \in Y^*$. Then:
(2)
\begin{align} \quad (T + S)^*(f) = f \circ (T + S) = f \circ T + f \circ S = T^*(f) + S^*(f) \quad \blacksquare \end{align}
• Proof of b) Let $f \in Y^*$ and $k \in \mathbb{R}$. Then:
(3)
\begin{align} \quad (kT)^* = f \circ (kT) = k (f \circ T) = kT^*(f) \quad \blacksquare \end{align}