In functional analysis, an unbounded operator is a linear map between infinite‑dimensional normed spaces (most commonly between Hilbert or Banach spaces) that is not bounded; equivalently, its operator norm is infinite. Because unbounded operators cannot be defined on the whole space without losing linearity or continuity, they are typically studied as densely defined operators, i.e., linear maps whose domain is a dense linear subspace of the source space.
Formal definition
Let $X$ and $Y$ be normed linear spaces.
A linear operator $T\colon D(T)\subseteq X\to Y$ is called unbounded if
$$ \sup_{\substack{x\in D(T)\ |x|\le 1}} |Tx| = \infty . $$
If the supremum is finite, $T$ is bounded and extends uniquely to a continuous linear operator on the closure of its domain.
Because any continuous linear operator on a Banach space is bounded (the Closed Graph Theorem), unbounded operators can exist only when the domain $D(T)$ is a proper subspace of $X$. The usual setting is that $D(T)$ is dense in $X$, which allows the use of adjoints, closures, and spectral theory.
Basic properties
| Property | Description |
|---|---|
| Domain | A linear subspace $D(T)\subseteq X$; for most theory one assumes $D(T)$ is dense in $X$. |
| Closedness | An operator $T$ is closed if its graph ${(x,Tx): x\in D(T)}$ is a closed subset of $X\times Y$. Many important unbounded operators are closed or closable. |
| Closability | $T$ is closable if it has a closed extension, i.e., the closure of its graph is the graph of another operator $\overline{T}$. |
| Adjoint | For densely defined $T\colon D(T)\subseteq H\to H$ on a Hilbert space $H$, the adjoint $T^{}$ is defined on the set of $y\in H$ such that the functional $x\mapsto\langle Tx,y\rangle$ extends continuously to all of $H$. $T^{}$ is always closed. |
| Self‑adjointness | A densely defined closed operator $T$ satisfies $T=T^{*}$. Self‑adjoint operators are central in quantum mechanics and spectral theory. |
| Symmetric | $T$ is symmetric when $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all $x,y\in D(T)$. Every self‑adjoint operator is symmetric, but the converse need not hold. |
| Essential self‑adjointness | A symmetric operator whose closure is self‑adjoint. |
Examples
-
Differentiation on $L^{2}(\mathbb{R})$
$$ (Df)(x)=f'(x),\qquad D(D)={f\in L^{2}(\mathbb{R})\mid f\text{ absolutely continuous},, f'\in L^{2}(\mathbb{R})}. $$ The operator $D$ is unbounded, densely defined, closed, and skew‑adjoint (i.e., $iD$ is self‑adjoint). -
Multiplication by the independent variable
On $L^{2}(\mathbb{R})$, define $(Mf)(x)=x f(x)$ with domain $D(M)={f\in L^{2}(\mathbb{R})\mid x f(x)\in L^{2}(\mathbb{R})}$. $M$ is self‑adjoint and unbounded. -
Momentum operator in quantum mechanics
In one dimension, $P = -i\hbar \frac{d}{dx}$ acting on $L^{2}(\mathbb{R})$ with the same domain as the differentiation operator above. It is essentially self‑adjoint. -
Laplacian on $L^{2}(\Omega)$
For a bounded open set $\Omega\subset\mathbb{R}^{n}$ with suitable boundary conditions (e.g., Dirichlet), the Laplacian $\Delta$ defined on $H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ is an unbounded, self‑adjoint, non‑negative operator.
Spectral theory
The spectral theorem extends to unbounded self‑adjoint operators: such an operator $A$ can be represented as an integral over its spectrum with respect to a projection‑valued measure $E$,
$$ A = \int_{\sigma(A)} \lambda , dE(\lambda). $$
This formulation underlies functional calculus for unbounded operators and is fundamental in the mathematical formulation of quantum mechanics.
Graph norm
For a densely defined operator $T$, the graph norm on $D(T)$ is defined by
$$ |x|_{T} = \bigl(|x|^{2} + |Tx|^{2}\bigr)^{1/2}. $$
With this norm, $D(T)$ becomes a Banach (indeed Hilbert, if $X$ is Hilbert) space precisely when $T$ is closed. The graph norm is useful for discussing convergence of sequences in the domain of an unbounded operator.
Key theorems
- Closed Graph Theorem (Banach spaces): A linear operator defined on all of a Banach space is bounded iff its graph is closed. Hence an everywhere‑defined linear operator on a Banach space cannot be unbounded.
- Hellinger–Toeplitz Theorem: Any symmetric operator defined on the whole Hilbert space is bounded; consequently, non‑trivial symmetric operators must have proper dense domains.
- Kato–Rellich Theorem: If $A$ is self‑adjoint and $B$ is symmetric and $A$-bounded with relative bound less than 1, then $A+B$ is self‑adjoint on $D(A)$. This result is frequently used to construct unbounded operators as perturbations of simpler ones.
Applications
- Quantum mechanics: Observables (position, momentum, energy) are modeled by self‑adjoint unbounded operators on $L^{2}$ spaces.
- Partial differential equations: Elliptic operators (Laplacian, Dirichlet‑Neumann operators) are unbounded and generate semigroups that solve evolution equations.
- Semigroup theory: The Hille‑Yosida theorem characterizes generators of strongly continuous one‑parameter semigroups; generators are closed, densely defined, typically unbounded operators.
References
- R. G. Douglas, Banach Algebra Techniques in Operator Theory, Springer, 1998.
- M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1980.
- J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990.
- M. Reed, Functional Analysis, 2nd ed., Academic Press, 1980.
These sources provide rigorous development of the theory of unbounded operators, their spectral properties, and applications.
