Chapter 4

The Adjoint Operator

The Hilbert space structure lets every bounded operator $T$ be paired with a canonical partner $T^*$ — the operation that turns $B(H)$ into a $C^*$-algebra.

Existence via the Riesz representation theorem

The construction of the adjoint rests on a foundational fact about Hilbert spaces:

Riesz representation theorem For every bounded linear functional $\varphi : H \to \mathbb{C}$, there is a unique $y \in H$ such that $\varphi(x) = \langle x, y \rangle$ for all $x \in H$, and $\|\varphi\| = \|y\|$.

Given $T \in B(H)$ and fixing $y \in H$, the map $x \mapsto \langle Tx, y \rangle$ is a bounded linear functional of $x$ (bounded because $|\langle Tx, y\rangle| \le \|T\|\|x\|\|y\|$). By Riesz representation there is a unique vector, which we call $T^*y$, satisfying $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x$. One checks $y \mapsto T^*y$ is linear and bounded, giving:

Definition The adjoint of $T \in B(H)$ is the unique operator $T^* \in B(H)$ satisfying $$ \langle Tx, y \rangle = \langle x, T^*y \rangle \qquad \text{for all } x, y \in H. $$

Algebraic properties

For $S, T \in B(H)$ and $\alpha \in \mathbb{C}$:

The $C^*$-identity

Theorem For every $T \in B(H)$, $$ \|T^*\| = \|T\| \qquad \text{and} \qquad \|T^*T\| = \|T\|^2. $$

Sketch. $\|T^*\| = \|T\|$ follows from Cauchy–Schwarz applied to $\langle Tx, y\rangle = \langle x, T^*y \rangle$, taking suprema over unit vectors on each side and using symmetry $(T^*)^*=T$. The identity $\|T^*T\| = \|T\|^2$ follows since $$ \|Tx\|^2 = \langle Tx, Tx \rangle = \langle x, T^*Tx \rangle \le \|x\|\,\|T^*Tx\| \le \|x\|^2 \|T^*T\|, $$ giving $\|T\|^2 \le \|T^*T\| \le \|T^*\|\|T\| = \|T\|^2$, so both inequalities are equalities.

This last identity — not just submultiplicativity, but exact equality $\|T^*T\| = \|T\|^2$ — is what makes $B(H)$ a $C^*$-algebra, and it is the algebraic seed from which most of operator theory grows.

Kernel, range, and the adjoint

Theorem For $T \in B(H)$, $$ \ker(T^*) = \operatorname{ran}(T)^{\perp}, \qquad \overline{\operatorname{ran}(T)} = \ker(T^*)^{\perp}. $$

This duality between kernels and (closures of) ranges under the adjoint is used repeatedly in spectral theory: it is, for instance, exactly what is needed to show that the spectrum of a self-adjoint operator is real (Chapter 6).