🧐Functional Analysis Unit 11 – Unbounded and Closed Linear Operators

Key Concepts and Definitions

• Linear operators map elements from one vector space to another while preserving linear combinations
• Unbounded linear operators have domains that are proper subsets of the entire vector space
• Closed linear operators have graphs that are closed subspaces of the product space
  • The graph of an operator $T$ is the set of all pairs $(x, Tx)$ where $x$ is in the domain of $T$
• The domain of an operator is the set of all vectors for which the operator is defined
• The range of an operator is the set of all vectors that are outputs of the operator
• Densely defined operators have domains that are dense in the underlying vector space
• The adjoint of an operator $T$ is the unique operator $T^*$ satisfying $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x$ in the domain of $T$ and $y$ in the domain of $T^*$

Unbounded Linear Operators

• Unbounded linear operators are not necessarily continuous and may not be defined on the entire vector space
• The domain of an unbounded operator is a proper subset of the vector space
• Unbounded operators can have unbounded norms, meaning $\|Tx\|$ can be arbitrarily large even for $x$ with $\|x\| = 1$
• Examples of unbounded operators include differentiation and integration operators on function spaces
  • The differentiation operator $Df = f'$ is unbounded on $C[0,1]$ with the supremum norm
• Unbounded operators may not have well-defined adjoints or inverses
• The spectrum of an unbounded operator can be more complicated than that of a bounded operator
  • It may include a continuous spectrum or residual spectrum in addition to point spectrum

Closed Linear Operators

• A linear operator $T$ is closed if its graph $G(T) = \{(x, Tx) : x \in D(T)\}$ is a closed subspace of $X \times Y$
  • Equivalently, if $x_n \to x$ and $Tx_n \to y$, then $x \in D(T)$ and $Tx = y$
• Closed operators have the property that convergence in the domain implies convergence in the range
• The domain of a closed operator is not necessarily dense in the vector space
• The sum and composition of closed operators may not be closed
• The adjoint of a densely defined closed operator is also closed
• Examples of closed operators include unbounded self-adjoint operators on Hilbert spaces
  • The momentum operator $-i\frac{d}{dx}$ on $L^2(\mathbb{R})$ with domain $H^1(\mathbb{R})$ is closed

Domains and Ranges

• The domain $D(T)$ of an operator $T$ is the set of all vectors $x$ for which $Tx$ is defined
  • For unbounded operators, $D(T)$ is a proper subset of the vector space
• The range $R(T)$ of an operator $T$ is the set of all vectors $y$ such that $y = Tx$ for some $x \in D(T)$
• The kernel or null space $N(T)$ is the set of all $x \in D(T)$ such that $Tx = 0$
• The graph of an operator $T$ is the set $G(T) = \{(x, Tx) : x \in D(T)\}$
  • For closed operators, $G(T)$ is a closed subspace of $X \times Y$
• The inverse $T^{-1}$ of an operator $T$ has domain $R(T)$ and range $D(T)$
  • Unbounded operators may not have well-defined inverses

Examples and Applications

• Differential operators on function spaces are common examples of unbounded operators
  • The first derivative operator $Df = f'$ on $C^1[0,1]$ is unbounded and closed
  • The Laplace operator $\Delta f = f''$ on $C^2[0,1]$ is also unbounded and closed
• Integral operators can be unbounded or bounded depending on the kernel function
  • The Volterra operator $(Vf)(x) = \int_0^x f(t) dt$ is bounded on $L^2[0,1]$
  • The Hilbert transform $(Hf)(x) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{f(t)}{x-t} dt$ is unbounded on $L^2(\mathbb{R})$
• Quantum mechanical observables are represented by unbounded self-adjoint operators on Hilbert spaces
  • The position operator $(Qf)(x) = xf(x)$ and momentum operator $(Pf)(x) = -i\hbar f'(x)$ are unbounded and closed on suitable domains in $L^2(\mathbb{R})$
• Closed operators arise naturally when studying differential equations and evolution equations
  • The heat equation $\frac{\partial u}{\partial t} = \Delta u$ can be formulated as an abstract Cauchy problem $\frac{du}{dt} = Au$ where $A$ is a closed operator

Theorems and Properties

• The Hellinger-Toeplitz theorem states that a symmetric operator defined on a Hilbert space is bounded
  • Consequently, unbounded self-adjoint operators cannot be everywhere defined
• The closed graph theorem characterizes closed operators in terms of convergent sequences
  • An operator $T$ is closed if and only if for every sequence $x_n \to x$ with $Tx_n \to y$, we have $x \in D(T)$ and $Tx = y$
• The Hille-Yosida theorem gives conditions for an operator to generate a $C_0$-semigroup of bounded operators
  • This is useful for studying evolution equations and abstract Cauchy problems
• The spectral theorem for unbounded self-adjoint operators extends the familiar spectral theory for compact self-adjoint operators
  • It allows for a continuous spectrum and a more general functional calculus
• The Stone theorem relates self-adjoint operators to one-parameter unitary groups
  • If $A$ is self-adjoint, then $U(t) = e^{itA}$ defines a unitary group, and conversely

Relationship to Bounded Operators

• Bounded operators are a special case of closed operators
  • Every bounded operator defined on the entire space is closed
• Unbounded operators can be approximated by bounded operators in various ways
  • The Yosida approximation $T_\lambda = \lambda T(\lambda I - T)^{-1}$ defines a bounded operator for $\lambda > 0$ in the resolvent set of $T$
  • The Friedrichs extension extends a symmetric operator to a self-adjoint operator by enlarging the domain
• Functional calculus can be defined for unbounded operators using limits of bounded operators
  • For a self-adjoint operator $A$, $f(A)$ can be defined as a limit of $f(A_n)$ where $A_n$ are bounded self-adjoint approximations
• Spectrum and resolvent theory for unbounded operators generalizes the bounded case
  • The resolvent $(\lambda I - T)^{-1}$ is bounded for $\lambda$ in the resolvent set of $T$
  • Spectral measures and projections can be defined for self-adjoint operators

Exercises and Problem-Solving Techniques

• To show an operator is unbounded, find a sequence $x_n$ in the domain with $\|x_n\| = 1$ but $\|Tx_n\| \to \infty$
  • For the differentiation operator on $C[0,1]$, consider $x_n(t) = \sin(nt)$
• To show an operator is closed, use the definition directly or apply the closed graph theorem
  • For the Laplace operator on $C^2[0,1]$, if $f_n \to f$ and $f_n'' \to g$, then $f \in C^2$ and $f'' = g$ by uniform convergence
• To find the adjoint of an unbounded operator, first find a dense domain on which the adjoint acts
  • For the momentum operator $Pf = -i\hbar f'$ on $L^2(\mathbb{R})$, the adjoint acts on the Sobolev space $H^1(\mathbb{R})$
• To solve eigenvalue problems for unbounded operators, use techniques from ODE and PDE theory
  • For the Sturm-Liouville problem $-(pf')' + qf = \lambda wf$, apply boundary conditions and orthogonality of eigenfunctions
• To study evolution equations, use semigroup theory and the Hille-Yosida theorem
  • For the heat equation $\frac{\partial u}{\partial t} = \Delta u$, show that the Laplace operator generates a contraction semigroup on a suitable function space