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).

Comments

A continuous linear operator from one topological vector space into another maps bounded sets into bounded sets. The converse is also true for linear mappings between normed linear spaces.

References

[a1]	S. Goldberg, “Unbounded linear operators”, McGraw-Hill (1966).
[a2]	I.C. Gohberg, S. Goldberg, M.A. Kaashoek, “Classes of linear operators”, 1, Birkhäuser (1991).