closable operator
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In mathematics, more specifically
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
and
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
s in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, and other cases. The term "unbounded operator" can be misleading, since * "unbounded" should sometimes be understood as "not necessarily bounded"; * "operator" should be understood as " linear operator" (as in the case of "bounded operator"); * the domain of the operator is a linear subspace, not necessarily the whole space; * this linear subspace is not necessarily closed; often (but not always) it is assumed to be
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
; * in the special case of a bounded operator, still, the domain is usually assumed to be the whole space. In contrast to
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
s, unbounded operators on a given space do not form an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s are possible.


Short history

The theory of unbounded operators developed in the late 1920s and early 1930s as part of developing a rigorous mathematical framework for
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
. The theory's development is due to
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
and Marshall Stone. Von Neumann introduced using graphs to analyze unbounded operators in 1932.


Definitions and basic properties

Let be Banach spaces. An unbounded operator (or simply ''operator'') is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
from a linear subspace —the domain of —to the space . Contrary to the usual convention, may not be defined on the whole space . An operator is said to be closed if its
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
is a closed set. (Here, the graph is a linear subspace of the direct sum , defined as the set of all pairs , where runs over the domain of  .) Explicitly, this means that for every sequence of points from the domain of such that and , it holds that belongs to the domain of and . The closedness can also be formulated in terms of the ''graph norm'': an operator is closed if and only if its domain is a
complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
with respect to the norm: : \, x\, _T = \sqrt. An operator is said to be densely defined if its domain is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in . This also includes operators defined on the entire space , since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint (if and are Hilbert spaces) and the transpose; see the sections below. If is closed, densely defined and continuous on its domain, then its domain is all of . A densely defined operator on a Hilbert space is called bounded from below if is a positive operator for some real number . That is, for all in the domain of (or alternatively since is arbitrary). If both and are bounded from below then is bounded.


Example

Let denote the space of continuous functions on the unit interval, and let denote the space of continuously differentiable functions. We equip C( ,1 with the supremum norm, \, \cdot\, _, making it a Banach space. Define the classical differentiation operator by the usual formula: : \left (\fracf \right )(x) = \lim_ \frac, \qquad \forall x \in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
Every differentiable function is continuous, so . We claim that is a well-defined unbounded operator, with domain . For this, we need to show that \frac is linear and then, for example, exhibit some \_n \subset C^1( ,1 such that \, f_n\, _\infty=1 and \sup_n \, \frac f_n\, _\infty=+\infty. This is a linear operator, since a linear combination of two continuously differentiable functions is also continuously differentiable, and :\left (\tfrac \right )(af+bg)= a \left (\tfrac f \right ) + b \left (\tfrac g \right ). The operator is not bounded. For example, :\begin f_n :
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\to 1, 1\\ f_n(x) = \sin (2\pi n x) \end satisfy : \left \, f_n \right \, _ = 1, but : \left \, \left (\tfrac f_n \right ) \right \, _ = 2\pi n \to \infty as n\to\infty. The operator is densely defined, and closed. The same operator can be treated as an operator for many choices of Banach space and not be bounded between any of them. At the same time, it can be bounded as an operator for other pairs of Banach spaces , and also as operator for some topological vector spaces . As an example let be an open interval and consider :\frac : \left (C^1 (I), \, \cdot \, _ \right ) \to \left ( C (I), \, \cdot \, _ \right), where: :\, f \, _ = \, f \, _ + \, f' \, _.


Adjoint

The adjoint of an unbounded operator can be defined in two equivalent ways. Let T : D(T) \subseteq H_1 \to H_2 be an unbounded operator between Hilbert spaces. First, it can be defined in a way analogous to how one defines the adjoint of a bounded operator. Namely, the adjoint T^* : D\left(T^*\right) \subseteq H_2 \to H_1 of is defined as an operator with the property: \langle Tx \mid y \rangle_2 = \left \langle x \mid T^*y \right \rangle_1, \qquad x \in D(T). More precisely, T^* y is defined in the following way. If y \in H_2 is such that x \mapsto \langle Tx \mid y \rangle is a continuous linear functional on the domain of , then y is declared to be an element of D\left(T^*\right), and after extending the linear functional to the whole space via the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
, it is possible to find some z in H_1 such that \langle Tx \mid y \rangle_2 = \langle x \mid z \rangle_1, \qquad x \in D(T), since
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, ...
allows the continuous dual of the Hilbert space H_1 to be identified with the set of linear functionals given by the inner product. This vector z is uniquely determined by y if and only if the linear functional x \mapsto \langle Tx \mid y \rangle is densely defined; or equivalently, if is densely defined. Finally, letting T^* y = z completes the construction of T^*, which is necessarily a linear map. The adjoint T^* y exists if and only if is densely defined. By definition, the domain of T^* consists of elements y in H_2 such that x \mapsto \langle Tx \mid y \rangle is continuous on the domain of . Consequently, the domain of T^* could be anything; it could be trivial (that is, contains only zero). It may happen that the domain of T^* is a closed hyperplane and T^* vanishes everywhere on the domain. Thus, boundedness of T^* on its domain does not imply boundedness of . On the other hand, if T^* is defined on the whole space then is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. If the domain of T^* is dense, then it has its adjoint T^. A closed densely defined operator is bounded if and only if T^* is bounded. The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator J as follows: \begin J: H_1 \oplus H_2 \to H_2 \oplus H_1 \\ J(x \oplus y) = -y \oplus x \end Since J is an isometric surjection, it is unitary. Hence: J(\Gamma(T))^ is the graph of some operator S if and only if is densely defined. A simple calculation shows that this "some" S satisfies: \langle Tx \mid y \rangle_2 = \langle x \mid Sy \rangle_1, for every in the domain of . Thus S is the adjoint of . It follows immediately from the above definition that the adjoint T^* is closed. In particular, a self-adjoint operator (meaning T = T^*) is closed. An operator is closed and densely defined if and only if T^ = T. Some well-known properties for bounded operators generalize to closed densely defined operators. The kernel of a closed operator is closed. Moreover, the kernel of a closed densely defined operator T : H_1 \to H_2 coincides with the orthogonal complement of the range of the adjoint. That is, \operatorname(T) = \operatorname(T^*)^\bot.
von Neumann's theorem In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces. Statement of the theorem Let G and H be Hilbert spaces, and let T : \operatorname(T) \subseteq G \to H be an unbounded operator from G ...
states that T^* T and T T^* are self-adjoint, and that I + T^* T and I + T T^* both have bounded inverses. If T^* has trivial kernel, has dense range (by the above identity.) Moreover: : is surjective if and only if there is a K > 0 such that \, f\, _2 \leq K \left\, T^* f\right\, _1 for all f in D\left(T^*\right). (This is essentially a variant of the so-called closed range theorem.) In particular, has closed range if and only if T^* has closed range. In contrast to the bounded case, it is not necessary that (T S)^* = S^* T^*, since, for example, it is even possible that (T S)^* does not exist. This is, however, the case if, for example, is bounded. A densely defined, closed operator is called ''
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
'' if it satisfies the following equivalent conditions: * T^* T = T T^*; * the domain of is equal to the domain of T^*, and \, T x\, = \left\, T^* x\right\, for every in this domain; * there exist self-adjoint operators A, B such that T = A + i B,T^* = A - i B, and \, T x\, ^2 = \, A x\, ^2 + \, B x\, ^2 for every in the domain of . Every self-adjoint operator is normal.


Transpose

Let T : B_1 \to B_2 be an operator between Banach spaces. Then the ''
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
'' (or ''dual'') ^t T: ^* \to ^* of T is the linear operator satisfying: \langle T x, y' \rangle = \langle x, \left(^t T\right) y' \rangle for all x \in B_1 and y \in B_2^*. Here, we used the notation: \langle x, x' \rangle = x'(x). The necessary and sufficient condition for the transpose of T to exist is that T is densely defined (for essentially the same reason as to adjoints, as discussed above.) For any Hilbert space H, there is the anti-linear isomorphism: J: H^* \to H given by J f = y where f(x) = \langle x \mid y \rangle_H, (x \in H). Through this isomorphism, the transpose ^t T relates to the adjoint T^* in the following way: T^* = J_1 \left(^t T\right) J_2^, where J_j: H_j^* \to H_j. (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose.


Closed linear operators

Closed linear operators are a class of linear operators on Banach spaces. They are more general than
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
s, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
and (with certain assumptions)
functional calculus In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral the ...
for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
and a large class of differential operators. Let be two Banach spaces. A linear operator is closed if for every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
in converging to in such that as one has and . Equivalently, is closed if its
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
is closed in the direct sum . Given a linear operator , not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of , and we say that is closable. Denote the closure of by . It follows that is the
restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and logi ...
of to . A core (or essential domain) of a closable operator is a subset of such that the closure of the restriction of to is .


Example

Consider the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
operator where is the Banach space of all continuous functions on an interval . If one takes its domain to be , then is a closed operator which is not bounded. On the other hand if , then will no longer be closed, but it will be closable, with the closure being its extension defined on .


Symmetric operators and self-adjoint operators

An operator ''T'' on a Hilbert space is ''symmetric'' if and only if for each ''x'' and ''y'' in the domain of we have \langle Tx \mid y \rangle = \lang x \mid Ty \rang. A densely defined operator is symmetric if and only if it agrees with its adjoint ''T'' restricted to the domain of ''T'', in other words when ''T'' is an extension of . In general, if ''T'' is densely defined and symmetric, the domain of the adjoint ''T'' need not equal the domain of ''T''. If ''T'' is symmetric and the domain of ''T'' and the domain of the adjoint coincide, then we say that ''T'' is ''self-adjoint''. Note that, when ''T'' is self-adjoint, the existence of the adjoint implies that ''T'' is densely defined and since ''T'' is necessarily closed, ''T'' is closed. A densely defined operator ''T'' is ''symmetric'', if the subspace (defined in a previous section) is orthogonal to its image under ''J'' (where ''J''(''x'',''y''):=(''y'',-''x'')).Follows from and the definition via adjoint operators. Equivalently, an operator ''T'' is ''self-adjoint'' if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators , are surjective, that is, map the domain of ''T'' onto the whole space ''H''. In other words: for every ''x'' in ''H'' there exist ''y'' and ''z'' in the domain of ''T'' such that and . An operator ''T'' is ''self-adjoint'', if the two subspaces , are orthogonal and their sum is the whole space H \oplus H . This approach does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators. A symmetric operator is often studied via its
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
. An operator ''T'' on a complex Hilbert space is symmetric if and only if its quadratic form is real, that is, the number \langle Tx \mid x \rangle is real for all ''x'' in the domain of ''T''. A densely defined closed symmetric operator ''T'' is self-adjoint if and only if ''T'' is symmetric. It may happen that it is not. A densely defined operator ''T'' is called ''positive'' (or ''nonnegative'') if its quadratic form is nonnegative, that is, \langle Tx \mid x \rangle \ge 0 for all ''x'' in the domain of ''T''. Such operator is necessarily symmetric. The operator ''T''''T'' is self-adjoint and positive for every densely defined, closed ''T''. The
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
applies to self-adjoint operators and moreover, to normal operators, but not to densely defined, closed operators in general, since in this case the spectrum can be empty. A symmetric operator defined everywhere is closed, therefore bounded, which is the Hellinger–Toeplitz theorem.


Extension-related

By definition, an operator ''T'' is an ''extension'' of an operator ''S'' if . An equivalent direct definition: for every ''x'' in the domain of ''S'', ''x'' belongs to the domain of ''T'' and . Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at Discontinuous linear map#General existence theorem and based on the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. If the given operator is not bounded then the extension is a
discontinuous linear map In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved ar ...
. It is of little use since it cannot preserve important properties of the given operator (see below), and usually is highly non-unique. An operator ''T'' is called ''closable'' if it satisfies the following equivalent conditions: * ''T'' has a closed extension; * the closure of the graph of ''T'' is the graph of some operator; * for every sequence (''xn'') of points from the domain of ''T'' such that ''xn'' → 0 and also ''Txn'' → ''y'' it holds that . Not all operators are closable. A closable operator ''T'' has the least closed extension \overline T called the ''closure'' of ''T''. The closure of the graph of ''T'' is equal to the graph of \overline T. Other, non-minimal closed extensions may exist. A densely defined operator ''T'' is closable if and only if ''T'' is densely defined. In this case \overline T = T^ and (\overline T)^* = T^*. If ''S'' is densely defined and ''T'' is an extension of ''S'' then ''S'' is an extension of ''T''. Every symmetric operator is closable. A symmetric operator is called ''maximal symmetric'' if it has no symmetric extensions, except for itself. Every self-adjoint operator is maximal symmetric. The converse is wrong. An operator is called ''essentially self-adjoint'' if its closure is self-adjoint. An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension. A symmetric operator may have more than one self-adjoint extension, and even a continuum of them. A densely defined, symmetric operator ''T'' is essentially self-adjoint if and only if both operators , have dense range. Let ''T'' be a densely defined operator. Denoting the relation "''T'' is an extension of ''S''" by ''S'' ⊂ ''T'' (a conventional abbreviation for Γ(''S'') ⊆ Γ(''T'')) one has the following. * If ''T'' is symmetric then ''T'' ⊂ ''T''∗∗ ⊂ ''T''. * If ''T'' is closed and symmetric then ''T'' = ''T''∗∗ ⊂ ''T''. * If ''T'' is self-adjoint then ''T'' = ''T''∗∗ = ''T''. * If ''T'' is essentially self-adjoint then ''T'' ⊂ ''T''∗∗ = ''T''.


Importance of self-adjoint operators

The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
holds for self-adjoint operators. In combination with
Stone's theorem on one-parameter unitary groups In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families :(U_)_ o ...
it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see Self-adjoint operator#Self-adjoint extensions in quantum mechanics. Such unitary groups are especially important for describing time evolution in classical and quantum mechanics.


See also

* Hilbert space#Unbounded operators * Stone–von Neumann theorem *
Bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...


Notes


References

* (see Chapter 12 "General theory of unbounded operators in Hilbert spaces"). * * * * * (see Chapter 5 "Unbounded operators"). * (see Chapter 8 "Unbounded operators"). * * {{DEFAULTSORT:Unbounded Operator Linear operators Operator theory de:Linearer Operator#Unbeschränkte lineare Operatoren