8.3 Spectral theorem for normal operators

Normal operators are a crucial class of linear operators in Hilbert spaces. They include self-adjoint and unitary operators, which are fundamental in quantum mechanics. Understanding normal operators is key to grasping the structure of linear transformations in complex vector spaces.

The spectral theorem for normal operators is a powerful tool in functional analysis. It allows us to decompose these operators into simpler parts, making it easier to study their properties and applications in various fields of mathematics and physics.

Normal Operators and the Spectral Theorem

Normal operators and examples

• Normal operators defined as linear operators on a Hilbert space that commute with their adjoint ($T^*T = TT^*$)
• Self-adjoint operators are normal operators equal to their own adjoint ($T = T^*$)
  • Position operator $Q$ and momentum operator $P$ in quantum mechanics are self-adjoint
• Unitary operators are normal operators whose adjoint is their inverse ($U^*U = UU^* = I$)
  • Preserve inner products and norms of vectors in the Hilbert space
  • Time-evolution operator $U(t) = e^{-iHt/\hbar}$ in quantum mechanics is unitary ($H$ is the Hamiltonian)

Spectral theorem for compact normals

• Compact normal operators on a Hilbert space $H$ possess an orthonormal basis $\{e_n\}$ consisting of their eigenvectors
• Corresponding eigenvalues $\{\lambda_n\}$ of the compact normal operator converge to zero
• Proof outline:
  1. Eigenvectors corresponding to distinct eigenvalues are orthogonal
  2. Eigenspaces of the compact normal operator span the entire Hilbert space $H$
  3. Eigenvalues converge to zero due to the compactness of the operator

Extension to bounded normals

• Bounded normal operators on a Hilbert space $H$ have a unique resolution of the identity $E$
• Spectral decomposition: $T = \int_{\sigma(T)} \lambda dE(\lambda)$, where $\sigma(T)$ is the spectrum of $T$
• Functional calculus extends the spectral theorem to bounded normal operators
  • For a bounded Borel function $f$ on $\sigma(T)$, the operator $f(T) = \int_{\sigma(T)} f(\lambda) dE(\lambda)$
  • Functional calculus properties: $(fg)(T) = f(T)g(T)$, $(\overline{f})(T) = f(T)^*$, $\|f(T)\| \leq \sup_{\lambda \in \sigma(T)} |f(\lambda)|$

Applications of spectral theorem

• Diagonalization of normal operators
  • Normal operators are unitarily equivalent to multiplication operators
  • Unitary operator $U$ exists such that $T = U^*MU$, where $M$ is the multiplication operator $Me_n = \lambda_n e_n$
• Computing spectra of normal operators
  • Spectrum of a normal operator consists of its eigenvalues
  • Compact normal operators have a countable spectrum with the only accumulation point at zero
  • Bounded normal operators have a compact spectrum in the complex plane
• Quantum mechanics applications
  • Observables are self-adjoint operators with real spectra and complete orthonormal eigenvectors
  • Measurements are projections onto eigenvectors of the observable's self-adjoint operator
• Diagonalization of the unilateral shift operator $S$ on $\ell^2(\mathbb{N})$
  • $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$ is a bounded normal operator
  • Spectrum of $S$ is $\sigma(S) = \{\lambda \in \mathbb{C} : |\lambda| \leq 1\}$
  • Spectral theorem diagonalizes $S$ as a multiplication operator on a suitable Hilbert space