11.4 Spectral theory for unbounded self-adjoint operators

Unbounded self-adjoint operators are crucial in functional analysis, extending beyond bounded operators. They're defined on dense subspaces of Hilbert spaces and have unique spectral properties, making them essential for understanding complex mathematical structures.

The spectral theorem for these operators provides a powerful tool for analysis. It allows for operator decomposition, domain characterization, and diagonalization, enabling applications in quantum mechanics, differential equations, and operator algebras.

Spectral Theory for Unbounded Self-Adjoint Operators

Self-adjoint unbounded operators

• Unbounded operators are linear operators $T: D(T) \subset H \to H$ where the domain $D(T)$ is a dense subspace of the Hilbert space $H$
  • Not necessarily bounded or continuous, unlike bounded operators defined on the entire Hilbert space
• Self-adjoint unbounded operators satisfy $T = T^*$, where $T^*$ is the adjoint of $T$
  • Domain of $T$ equals the domain of its adjoint: $D(T) = D(T^*)$, ensuring the operator is well-defined
  • Inner product equality: $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in D(T)$, expressing symmetry
• Spectrum of a self-adjoint unbounded operator consists of:
  • Point spectrum: eigenvalues $\lambda$ satisfying $Tx = \lambda x$ for some $x \in D(T)$
  • Continuous spectrum: $\lambda \in \mathbb{C}$ such that $T - \lambda I$ is not surjective, but its range is dense in $H$
  • Residual spectrum is empty for self-adjoint operators, unlike non-self-adjoint operators

Spectral theorem for unbounded operators

• The spectral theorem for unbounded self-adjoint operators guarantees the existence of a unique spectral measure $E$ on the Borel $\sigma$-algebra of $\mathbb{R}$
  • Operator decomposition: $T = \int_{\mathbb{R}} \lambda dE(\lambda)$, expressing $T$ as an integral with respect to the spectral measure
  • Domain characterization: $D(T) = \left\{ x \in H : \int_{\mathbb{R}} |\lambda|^2 d\langle E(\lambda)x, x \rangle < \infty \right\}$, describing the domain of $T$ using the spectral measure
• Interpretation of the spectral theorem:
  • The spectral measure $E$ decomposes the Hilbert space $H$ into a direct integral of Hilbert spaces $H_{\lambda}$ corresponding to the spectral values $\lambda$
  • The operator $T$ acts as multiplication by $\lambda$ on each $H_{\lambda}$, simplifying its action
  • The spectral theorem provides a way to diagonalize unbounded self-adjoint operators, generalizing the concept of eigendecomposition

Diagonalization of unbounded operators

• Diagonalization using the spectral theorem involves defining a unitary operator $U: H \to L^2(\mathbb{R}, \mu)$:
  1. $(Ux)(\lambda) = \frac{d\langle E(\lambda)x, x \rangle}{d\mu(\lambda)}$, mapping elements of $H$ to functions in $L^2(\mathbb{R}, \mu)$
  2. $\mu$ is the scalar-valued spectral measure given by $\mu(B) = \langle E(B)x, x \rangle$ for Borel sets $B \subset \mathbb{R}$
• The operator $T$ is unitarily equivalent to the multiplication operator $M_{\lambda}$ on $L^2(\mathbb{R}, \mu)$:
  • $UTU^{-1} = M_{\lambda}$, where $(M_{\lambda}f)(\lambda) = \lambda f(\lambda)$, demonstrating the diagonalization of $T$
• Functional calculus allows the construction of new operators from the spectral measure of $T$:
  • For a bounded Borel function $f: \mathbb{R} \to \mathbb{C}$, define the operator $f(T)$ by $f(T) = \int_{\mathbb{R}} f(\lambda) dE(\lambda)$
  • Functional calculus extends the notion of applying functions to operators, enabling the study of operator properties

Applications of unbounded operator theory

• Quantum mechanics heavily relies on self-adjoint operators to represent observables
  • The spectral theorem allows the decomposition of the Hilbert space into eigenspaces corresponding to possible measurement outcomes
  • Expectation values and probabilities can be calculated using the spectral measure, providing a probabilistic interpretation
• Differential operators, such as the Laplace operator or the Schrödinger operator, are often unbounded self-adjoint operators
  • The spectral theorem can be used to analyze the spectrum and eigenfunctions of these operators
  • Spectral properties of differential operators have applications in physics (quantum mechanics, wave equations), engineering (vibration analysis), and partial differential equations (existence and uniqueness of solutions)
• Functional calculus and operator algebras benefit from the spectral theorem
  • The functional calculus allows the construction of new operators from a given self-adjoint operator, enriching the operator algebra
  • The spectral theorem provides a powerful tool for analyzing the structure of operator algebras generated by self-adjoint operators, such as von Neumann algebras and C*-algebras