Worked Examples

Four concrete operators, computed explicitly, that illustrate every notion introduced so far: norm, adjoint, normality, and spectrum.

1. The right shift on $\ell^2$

• Bounded: $\|Sa\|^2 = \sum_{n\ge1} |a_n|^2 = \|a\|^2$, so $S$ is an isometry and $\|S\| = 1$.
• Adjoint: a direct computation with $\langle Sa, b\rangle = \sum_{n \ge 1} a_n \overline{b_{n+1}}$ shows $$ S^*(b_1, b_2, b_3, \dots) = (b_2, b_3, b_4, \dots) \qquad \text{(the left shift).} $$
• Not normal: $S^*S = I$ (since $S$ is an isometry) but $SS^*(a_1, a_2, \dots) = (0, a_2, a_3, \dots) \ne I$, so $SS^* \ne S^*S$. $S$ is an isometry but not unitary — it is injective but not surjective (its range misses vectors with nonzero first coordinate).
• Spectrum: $\sigma(S) = \{ \lambda \in \mathbb{C} : |\lambda| \le 1 \}$, the closed unit disk. $S$ has no eigenvalues at all ($Sa = \lambda a$ forces $a = 0$ by comparing coordinates), so the entire spectrum consists of continuous and residual spectrum — a purely infinite-dimensional phenomenon.

2. A diagonal (multiplication) operator on $\ell^2$

• Norm: $\|D\| = \sup_n |\lambda_n|$ (derived in Chapter 3).
• Adjoint: $D^*(b_1, b_2, \dots) = (\overline{\lambda_1} b_1, \overline{\lambda_2} b_2, \dots)$ — also diagonal, with conjugated entries.
• Normal: $D^*D$ and $DD^*$ are both the diagonal operator with entries $|\lambda_n|^2$, so $D$ is always normal.
• Self-adjoint iff every $\lambda_n \in \mathbb{R}$; unitary iff every $|\lambda_n| = 1$.
• Spectrum: the point spectrum is $\{\lambda_n : n \ge 1\}$ (each $e_n$ is an eigenvector), and $\sigma(D)$ is the closure of this set.
• Compact iff $\lambda_n \to 0$ as $n \to \infty$ — a first, very concrete instance of the Riesz–Schauder picture from Chapter 7.

3. Multiplication operator on $L^2([0,1])$

• Bounded with $\|M_\varphi\| = \|\varphi\|_\infty$ (the essential supremum of $|\varphi|$).
• Adjoint: $M_\varphi^* = M_{\overline{\varphi}}$, so $M_\varphi$ is always normal, self-adjoint iff $\varphi$ is real-valued a.e., and unitary iff $|\varphi(t)| = 1$ a.e.
• Spectrum: $\sigma(M_\varphi)$ equals the essential range of $\varphi$ — the set of $\lambda$ such that $\{ t : |\varphi(t) - \lambda| < \varepsilon \}$ has positive measure for every $\varepsilon > 0$. If $\varphi$ is continuous and non-constant, this is exactly $\varphi([0,1])$, and $M_\varphi$ has no eigenvalues — the entire spectrum is continuous spectrum.

4. A Fredholm integral (Hilbert–Schmidt) operator

• Bounded and compact: $T$ is always compact — this is the classical fact that Hilbert–Schmidt integral operators are compact, with $\|T\| \le \left( \int_0^1\!\int_0^1 |k(t,s)|^2 \, dt \, ds \right)^{1/2}$.
• Adjoint: $T^* f(t) = \int_0^1 \overline{k(s,t)} f(s) \, ds$ — the integral operator with kernel $\overline{k(s,t)}$.
• Self-adjoint iff $k(t,s) = \overline{k(s,t)}$ for a.e. $(t,s)$ — a Hermitian kernel. In that case, the spectral theorem of Chapter 7 applies directly: $T$ has real eigenvalues $\lambda_n \to 0$ and an orthonormal basis of eigenfunctions, giving the eigenfunction expansion $Tf = \sum_n \lambda_n \langle f, e_n \rangle e_n$ used throughout the classical theory of integral equations.