In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the composition operator
with symbol
is a
linear operator
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 pre ...
defined by the rule
where
denotes
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
.
The study of composition operators is covered b
AMS category 47B33
In physics
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and especially the area of
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
, the composition operator is usually referred to as the Koopman operator (and its wild surge in popularity is sometimes jokingly called "Koopmania"), named after
Bernard Koopman
Bernard Osgood Koopman (January 19, 1900 – August 18, 1981) was a French-born American mathematician, known for his work in ergodic theory, the foundations of probability, statistical theory and operations research.
Education and work
Af ...
. It is the
left-adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
of the
transfer operator
Transfer may refer to:
Arts and media
* ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović
* ''Transfer'' (1966 film), a short film
* ''Transfer'' (journal), in management studies
...
of Frobenius–Perron.
In Borel functional calculus
Using the language of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, the composition operator is a
pull-back
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: in ...
on the space of
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
s; it is adjoint to the
transfer operator
Transfer may refer to:
Arts and media
* ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović
* ''Transfer'' (1966 film), a short film
* ''Transfer'' (journal), in management studies
...
in the same way that the pull-back is adjoint to the
push-forward; the composition operator is the
inverse image functor In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor ...
.
Since the domain considered here is that of
Borel function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s, the above describes the Koopman operator as it appears in
Borel functional calculus
In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. ...
.
In holomorphic functional calculus
The
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
of a composition operator can be taken more narrowly, as some
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, often consisting of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s: for example, some
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . ...
or
Bergman space In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain ''D'' of the complex plane that are sufficiently well-behaved at the boundary that t ...
. In this case, the composition operator lies in the realm of some
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 theo ...
, such as the
holomorphic functional calculus
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ''f'' of a complex argument ''z'' and an operator ''T'', the aim is to construct an operator, ''f''(''T ...
.
Interesting questions posed in the study of composition operators often relate to how the
spectral properties
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of the operator depend on the
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
. Other questions include whether
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
or
trace-class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace- ...
; answers typically depend on how the function
behaves on the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
of some domain.
When the transfer operator is a left-
shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift o ...
, the Koopman operator, as its adjoint, can be taken to be the right-shift operator. An appropriate basis, explicitly manifesting the shift, can often be found in the
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomial ...
. When these are orthogonal on the real number line, the shift is given by the
Jacobi operator
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel ...
. When the polynomials are orthogonal on some region of the complex plane (viz, in
Bergman space In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain ''D'' of the complex plane that are sufficiently well-behaved at the boundary that t ...
), the Jacobi operator is replaced by a
Hessenberg operator Hessenberg may refer to:
People:
* Gerhard Hessenberg (1874–1925), German mathematician
* Karl Hessenberg (1904–1959), German mathematician and engineer
*Kurt Hessenberg (1908–1994), German composer and professor at the Hochschule für Musik ...
.
Applications
In mathematics, composition operators commonly occur in the study of
shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift o ...
s, for example, in the
Beurling–Lax theorem In mathematics, the Beurling–Lax theorem is a theorem due to and which characterizes the shift-invariant subspaces of the Hardy space H^2(\mathbb,\mathbb). It states that each such space is of the form
: \theta H^2(\mathbb,\mathbb),
for so ...
and the
Wold decomposition
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states ...
. Shift operators can be studied as one-dimensional
spin lattice
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discr ...
s. Composition operators appear in the theory of
Aleksandrov–Clark measure In mathematics, Aleksandrov–Clark (AC) measures are specially constructed Measure theory, measures named after the two mathematicians, Alexei Borisovich Aleksandrov, A. B. Aleksandrov and Douglas Clark (mathematician), Douglas Clark, who discovere ...
s.
The
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
equation of the composition operator is
Schröder's equation
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function , find the function such that
Schröder's equation is an eigenvalue equation for the composition operator that send ...
, and the principal
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
is often called
Schröder's function or
Koenigs function In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representation as dilations of a univalent holomorphic ma ...
.
Composition operator has been used in data-driven techniques for dynamical systems by the use of
dynamic mode decomposition
Dynamic mode decomposition (DMD) is a dimensionality reduction algorithm developed by Peter Schmid in 2008.
Given a time series of data, DMD computes a set of modes each of which is associated with a fixed oscillation frequency and decay/growth r ...
algorithms, which approximate the modes and eigenvalues of the composition operator.
See also
*
*
Carleman linearization
In mathematics, Carleman linearization (or Carleman embedding) is a technique to transform a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system. It was introduced by the Swedish mathematician Torsten Carleman i ...
*
*
*
*
Dynamic mode decomposition
Dynamic mode decomposition (DMD) is a dimensionality reduction algorithm developed by Peter Schmid in 2008.
Given a time series of data, DMD computes a set of modes each of which is associated with a fixed oscillation frequency and decay/growth r ...
References
* C. C. Cowen and
B. D. MacCluer, ''Composition operators on spaces of analytic functions''. Studies in Advanced Mathematics. CRC Press, Boca Raton, Florida, 1995. xii+388 pp. .
*
J. H. Shapiro, ''Composition operators and classical function theory.'' Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. xvi+223 pp. .
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Dynamical systems
Functional analysis
Operator theory
Topological vector spaces