Chapter 5

Special Classes of Operators

Four families defined purely in terms of $T$ and $T^*$ — self-adjoint, unitary, normal, and projection operators — that account for most operators encountered in practice.

Self-adjoint operators

Definition $T \in B(H)$ is self-adjoint (or Hermitian) if $T^* = T$, equivalently $$ \langle Tx, y \rangle = \langle x, Ty \rangle \qquad \text{for all } x, y \in H. $$

Self-adjoint operators are the infinite-dimensional analogue of real symmetric (or Hermitian) matrices. Two immediate consequences:

Unitary operators

Definition $T \in B(H)$ is unitary if $T^*T = TT^* = I$, i.e. $T$ is invertible with $T^{-1} = T^*$.

Equivalently, $T$ is unitary iff $T$ is a surjective isometry: $\|Tx\| = \|x\|$ for all $x$, and $T$ is onto. Unitary operators are exactly the isomorphisms of Hilbert space that preserve the inner product, $\langle Tx, Ty \rangle = \langle x, y \rangle$, so they are the natural notion of "change of orthonormal basis" or symmetry of $H$.

Normal operators

Definition $T \in B(H)$ is normal if $T$ commutes with its own adjoint: $$ T^*T = TT^*. $$

Both self-adjoint and unitary operators are normal, but normal operators form a strictly larger class. Normality is exactly the condition needed for the spectral theorem: normal operators are precisely those that are "diagonalizable" with respect to a suitable spectral measure. A useful equivalent characterization on a Hilbert space:

Theorem $T$ is normal if and only if $\|Tx\| = \|T^*x\|$ for every $x \in H$.

Sketch. $\|Tx\|^2 - \|T^*x\|^2 = \langle T^*Tx, x\rangle - \langle TT^*x, x \rangle = \langle (T^*T - TT^*)x, x\rangle$, which vanishes for all $x$ exactly when $T^*T - TT^* = 0$ (using that a self-adjoint operator $A$ with $\langle Ax,x\rangle = 0$ for all $x$ must be $0$).

Orthogonal projections

Definition $P \in B(H)$ is an orthogonal projection if $$ P^2 = P \qquad \text{(idempotent)} \qquad \text{and} \qquad P^* = P \qquad \text{(self-adjoint)}. $$

Every orthogonal projection $P$ corresponds to a closed subspace $M = \operatorname{ran}(P)$, with $P$ acting as the identity on $M$ and as $0$ on $M^{\perp}$, and every closed subspace arises this way. Projections satisfy $0 \le \langle Px, x \rangle \le \|x\|^2$ and $\|P\| \in \{0, 1\}$ for $P \ne 0$.

Summary of inclusions

Remark Self-adjoint $\subset$ Normal, and Unitary $\subset$ Normal, but self-adjoint and unitary operators are generally different classes (a self-adjoint unitary operator satisfies $T = T^* = T^{-1}$, i.e. $T^2 = I$ — a reflection). Orthogonal projections are self-adjoint but only idempotent ones among self-adjoint operators.