# Unbounded operator

A mapping $A$ from a set $M$ in a topological vector space $X$ into a topological vector space $Y$ such that there is a bounded set $N \subseteq M$ whose image $A[N]$ is an unbounded set in $Y$.

The simplest example of an unbounded operator is the differentiation operator $\dfrac{\mathrm{d}}{\mathrm{d}{t}}$, defined on the set ${C^{1}}([a,b])$ of all continuously differentiable functions into the space $C([a,b])$ of all continuous functions on $a \leq t \leq b$, because the operator $\dfrac{\mathrm{d}}{\mathrm{d}{t}}$ takes the bounded set $\{ t \mapsto \sin(n t) \}_{n \in \mathbb{N}}$ to the unbounded set $\{ t \mapsto n \cos(n t) \}_{n \in \mathbb{N}}$. An unbounded operator $A$ is necessarily discontinuous at certain (and if $A$ is linear, at all) points of its domain of definition. An important class of unbounded operators is that of the closed operators, because they have a property that to some extent replaces continuity.

Let $A$ and $B$ be unbounded operators with domains of definition $D_{A}$ and $D_{B}$ respectively. If $D_{A} \cap D_{B} \neq \varnothing$, then on this intersection, we can define the operator $(\alpha A + \beta B)(x) \stackrel{\text{df}}{=} \alpha A(x) + \beta B(x)$, where $\alpha,\beta \in \mathbf{R}$ (or $\mathbf{C}$), and similarly, if $D_{A} \cap {A^{\leftarrow}}[D_{B}] \neq \varnothing$, then we can define the operator $(B A)(x) \stackrel{\text{df}}{=} B(A(x))$. In particular, in this way, the powers $A^{k}$, where $k \in \mathbb{N}$, of an unbounded operator $A$ are defined. An operator $B$ is said to be an extension of an operator $A$, written $B \supseteq A$, if and only if $D_{A} \subseteq D_{B}$ and $B(x) = A(x)$ for $x \in D_{A}$. For example, $B (A_{1} + A_{2}) \supseteq B A_{1} + B A_{2}$. Commutativity of two operators is usually treated for the case when one of them is bounded: An unbounded operator $A$ commutes with a bounded operator $B$ if and only if $B A \subseteq A B$.

For unbounded linear operators, the concept of an adjoint operator is (still) defined. Let $A$ be an unbounded operator defined on a set $D_{A}$ that is dense in a topological vector space $X$ and mapping into a topological vector space $Y$. If $X^{*}$ and $Y^{*}$ are the strong dual spaces to $X$ and $Y$ respectively, and if $D_{A^{*}}$ is the collection of all linear functionals $\phi \in Y^{*}$ for which there exists a linear functional $f \in X^{*}$ such that $\langle A(x),\phi \rangle = \langle x,f \rangle$ for all $x \in D_{A}$, then the correspondence $\phi \mapsto f$ determines an operator $A^{*}$ on $D_{A^{*}}$ (which may, however, consists of the zero element only) in $Y^{*}$, the so-called adjoint operator of $A$.

#### References

 [1] K. Yosida, “Functional analysis”, Springer (1980), pp. Chapt. 8, §1. [2] N. Dunford, J.T. Schwartz, “Linear operators. General theory”, 1, Interscience (1958). [3] F. Riesz, B. Szökefalvi-Nagy, “Functional analysis”, F. Ungar (1955). (Translated from French) [4] L.A. [L.A. Lyusternik] Ljusternik, “Elements of functional analysis”, Wiley & Hindustan Publ. Comp. (1974). (Translated from Russian) [5] J. von Neumann, “Mathematische Grundlagen der Quantenmechanik”, Dover, reprint (1943).