🎭Operator Theory Unit 6 – Unbounded Operators

Key Concepts and Definitions

• Unbounded operators are linear operators defined on a dense subspace of a Hilbert space that are not necessarily bounded or continuous
• The domain of an unbounded operator is a proper subspace of the Hilbert space, while the range may or may not be contained in the same space
• Unbounded operators can be closed or closable, depending on the properties of their graph (set of pairs $(x, Tx)$ where $x$ is in the domain and $Tx$ is the corresponding output)
• The adjoint of an unbounded operator $T$ is another unbounded operator $T^*$ defined on a dense subspace of the Hilbert space, satisfying $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x$ in the domain of $T$ and $y$ in the domain of $T^*$
  • The adjoint plays a crucial role in the study of self-adjoint and symmetric operators
• Spectral theory for unbounded operators involves the study of their spectrum (set of eigenvalues) and resolvent (operator-valued function that encodes information about the spectrum)
• Unbounded operators are essential in quantum mechanics, where they represent physical observables such as position, momentum, and energy

Domain and Range of Unbounded Operators

• The domain of an unbounded operator $T$ is a dense linear subspace $D(T)$ of the Hilbert space $H$, meaning that any element in $H$ can be approximated by elements in $D(T)$
• Elements in the domain must satisfy certain conditions to ensure that the operator can be applied to them and the result remains in the Hilbert space
• The range of an unbounded operator $R(T)$ is the set of all elements in the Hilbert space that can be obtained by applying the operator to elements in its domain
  • Unlike bounded operators, the range of an unbounded operator may not be the entire Hilbert space or even a closed subspace
• The graph of an unbounded operator $G(T)$ is the set of all pairs $(x, Tx)$ where $x$ is in the domain and $Tx$ is the corresponding output
  • The properties of the graph (such as closedness) determine important characteristics of the operator
• The kernel (or null space) of an unbounded operator $\ker(T)$ is the set of all elements in the domain that are mapped to zero by the operator
• Unbounded operators can be injective (one-to-one) or surjective (onto) depending on the properties of their domain, range, and kernel

Examples of Unbounded Operators

• The differential operator $\frac{d}{dx}$ on the space of continuously differentiable functions $C^1([a, b])$ with domain $D(T) = C^1([a, b])$ and defined by $Tf = f'$ for $f \in D(T)$
• The multiplication operator $Mf(x) = xf(x)$ on the space of square-integrable functions $L^2(\mathbb{R})$ with domain $D(M) = \{f \in L^2(\mathbb{R}) : xf(x) \in L^2(\mathbb{R})\}$
• The momentum operator $P = -i\hbar\frac{d}{dx}$ in quantum mechanics, acting on the space of square-integrable functions $L^2(\mathbb{R})$ with domain $D(P) = \{f \in L^2(\mathbb{R}) : f' \in L^2(\mathbb{R})\}$
• The Laplace operator $\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ on the space of twice continuously differentiable functions $C^2(\Omega)$ with domain $D(\Delta) = C^2(\Omega) \cap C_0(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^3$ and $C_0(\Omega)$ denotes the space of continuous functions vanishing on the boundary of $\Omega$
• The inverse operator $T^{-1}$ of a bijective unbounded operator $T$, which is also unbounded and has domain $D(T^{-1}) = R(T)$

Properties and Characteristics

• Unbounded operators are linear, meaning that for any $x, y$ in the domain and scalars $\alpha, \beta$, $T(\alpha x + \beta y) = \alpha Tx + \beta Ty$
• Unlike bounded operators, unbounded operators may not be defined on the entire Hilbert space and may not be continuous
• The graph of a closed operator is a closed subspace of the product Hilbert space $H \times H$, while the graph of a closable operator has a closure that is the graph of an unbounded operator called its closure
• An unbounded operator $T$ is called densely defined if its domain $D(T)$ is dense in the Hilbert space $H$
  • Densely defined operators allow for the existence of the adjoint operator and the application of the closed graph theorem
• Unbounded operators may have a continuous inverse, even if they are not continuous themselves
• The spectrum of an unbounded operator can be divided into the point spectrum (eigenvalues), continuous spectrum, and residual spectrum
  • The spectral properties of unbounded operators are crucial in the study of differential equations and quantum mechanics
• Unbounded operators can be symmetric (or Hermitian) if $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y$ in the domain, or self-adjoint if $T = T^*$ (equality of operators and domains)

Closed and Closable Operators

• An unbounded operator $T$ is closed if its graph $G(T)$ is a closed subspace of the product Hilbert space $H \times H$
  • Equivalently, for any sequence $\{x_n\}$ in the domain of $T$ converging to $x$ in $H$, if $\{Tx_n\}$ converges to $y$ in $H$, then $x$ is in the domain of $T$ and $Tx = y$
• Closed operators have well-defined spectral properties and are essential in the study of differential equations and quantum mechanics
• An unbounded operator $T$ is closable if the closure of its graph $\overline{G(T)}$ is the graph of another unbounded operator, called the closure of $T$ and denoted by $\overline{T}$
  • The closure of a closable operator extends the original operator to a larger domain while preserving its action on the original domain
• The adjoint of a densely defined operator is always closed
• The closure of a closable operator is the smallest closed extension of the original operator
• Closed operators satisfy the closed graph theorem: a closed operator defined everywhere on a Hilbert space is bounded

Adjoint of Unbounded Operators

• The adjoint of a densely defined unbounded operator $T$ is another unbounded operator $T^*$ defined on a dense subspace $D(T^*)$ of the Hilbert space, satisfying $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x \in D(T)$ and $y \in D(T^*)$
• The domain of the adjoint $D(T^*)$ consists of all $y \in H$ for which the linear functional $x \mapsto \langle Tx, y \rangle$ is bounded on $D(T)$
  • For each such $y$, there exists a unique $z \in H$ such that $\langle Tx, y \rangle = \langle x, z \rangle$ for all $x \in D(T)$, and we define $T^*y = z$
• The adjoint operator is always closed, even if the original operator is not
• An unbounded operator $T$ is called symmetric (or Hermitian) if $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in D(T)$, which implies that $T \subset T^*$ (inclusion of operators and domains)
• An unbounded operator $T$ is called self-adjoint if $T = T^*$, meaning that the operators and their domains are equal
  • Self-adjoint operators play a fundamental role in quantum mechanics, as they represent observable quantities and ensure real eigenvalues

Applications in Quantum Mechanics

• In quantum mechanics, physical observables are represented by self-adjoint unbounded operators acting on a Hilbert space of wavefunctions
• The position operator $Q$ and momentum operator $P$ are unbounded self-adjoint operators satisfying the canonical commutation relation $[Q, P] = QP - PQ = i\hbar I$, where $\hbar$ is the reduced Planck's constant and $I$ is the identity operator
  • The non-commutativity of these operators leads to the Heisenberg uncertainty principle, which limits the precision of simultaneous measurements of position and momentum
• The Hamiltonian operator $H$, which represents the total energy of a quantum system, is an unbounded self-adjoint operator that governs the time evolution of the system through the Schrödinger equation $i\hbar \frac{\partial}{\partial t} \psi(t) = H \psi(t)$
• Spectral theory for unbounded self-adjoint operators allows for the decomposition of the Hilbert space into eigenspaces corresponding to different energy levels, which is crucial for understanding the behavior of quantum systems
• The study of unbounded operators in quantum mechanics has led to the development of powerful mathematical tools, such as the spectral theorem for unbounded self-adjoint operators and the theory of operator algebras

Challenges and Advanced Topics

• The theory of unbounded operators is technically more challenging than that of bounded operators due to the lack of uniform boundedness and continuity
• The Hellinger-Toeplitz theorem states that a symmetric unbounded operator defined everywhere on a Hilbert space must be bounded, highlighting the need for proper domain considerations
• The distinction between symmetric and self-adjoint unbounded operators is crucial, as not all symmetric operators are self-adjoint (e.g., the momentum operator with domain $C_0^{\infty}(\mathbb{R})$)
  • The deficiency indices of a symmetric operator measure the dimensions of the spaces of square-integrable solutions to the equations $(T^* \pm iI)f = 0$ and determine whether the operator has self-adjoint extensions
• The spectral theorem for unbounded self-adjoint operators requires a more sophisticated approach compared to the bounded case, involving the use of projection-valued measures and the functional calculus
• The study of unbounded operators on Banach spaces leads to the theory of closed operators and the notion of the resolvent set, which is essential in the study of differential equations and evolution problems
• The theory of operator semigroups, particularly strongly continuous semigroups (C_0-semigroups), provides a framework for studying the time evolution of systems governed by unbounded operators, with applications in partial differential equations, quantum mechanics, and stochastic processes