11.1 Definitions and examples of unbounded operators

Unbounded linear operators are crucial in functional analysis, extending beyond the limitations of bounded operators. They operate on specific domains within Hilbert spaces, allowing for the representation of complex mathematical and physical phenomena.

These operators, like differentiation and multiplication, play key roles in quantum mechanics and partial differential equations. Understanding their properties, such as closedness and spectral characteristics, is essential for solving real-world problems in physics and engineering.

Unbounded Linear Operators

Unbounded linear operators and domains

• Linear operators not necessarily bounded or continuous
  • Domain is a proper subspace of the Hilbert space ($L^2([a, b])$)
  • Operator $T: D(T) \rightarrow H$ maps from domain $D(T)$ to Hilbert space $H$
• Satisfies linearity property for $x, y \in D(T)$ and scalars $\alpha, \beta$: $T(\alpha x + \beta y) = \alpha T(x) + \beta T(y)$
• No constant $M > 0$ exists such that $\|Tx\| \leq M\|x\|$ for all $x \in D(T)$, unbounded
• Closed if its graph $G(T) = \{(x, Tx) : x \in D(T)\}$ is a closed subspace of $H \times H$

Examples of unbounded operators

• Differentiation operator on $L^2([a, b])$
  • $D(T) = \{f \in L^2([a, b]) : f \text{ absolutely continuous and } f' \in L^2([a, b])\}$
  • $(Tf)(x) = f'(x)$ for $f \in D(T)$, $x \in [a, b]$
• Multiplication operator on $L^2(\mathbb{R})$
  • $D(T) = \{f \in L^2(\mathbb{R}) : xf(x) \in L^2(\mathbb{R})\}$
  • $(Tf)(x) = xf(x)$ for $f \in D(T)$, $x \in \mathbb{R}$
• Laplace operator on $L^2(\mathbb{R}^n)$
  • $D(T) = H^2(\mathbb{R}^n) = \{f \in L^2(\mathbb{R}^n) : \partial^\alpha f \in L^2(\mathbb{R}^n) \text{ for all } |\alpha| \leq 2\}$
  • $(Tf)(x) = -\Delta f(x) = -\sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2}(x)$ for $f \in D(T)$, $x \in \mathbb{R}^n$

Bounded vs unbounded operators

• Bounded operators
  • Defined on entire Hilbert space
  • Continuous, small input changes lead to small output changes
  • Finite operator norm $\|T\| = \sup\{\|Tx\| : \|x\| \leq 1\} < \infty$
• Unbounded operators
  • Defined on proper subspace (domain) of Hilbert space
  • Not necessarily continuous
  • No finite operator norm
• Bounded operators have compact spectrum, unbounded can have unbounded spectrum
• Bounded operators always have bounded adjoint, unbounded may have unbounded or non-existent adjoints

Importance in functional analysis

• Quantum mechanics models observables as unbounded self-adjoint operators (position, momentum, Hamiltonian)
• Differential operators in PDEs often unbounded, studying properties helps understand solution behavior
• Unbounded operators with compact resolvents have discrete spectrum, spectral decomposition leads to generalized eigenfunction expansions
• Infinitesimal generators of strongly continuous semigroups often unbounded, applied in evolution equations (heat equation, Schrödinger equation)