In
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 ...
, a multiplication operator is an operator defined on some
vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
for all in the
domain of , and all in the domain of (which is the same as the domain of ).
This type of operator is often contrasted with
composition operator
In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule
C_\phi (f) = f \circ \phi
where f \circ \phi denotes function composition.
The study of composition operators is covered bAMS category 47B33 ...
s. Multiplication operators generalize the notion of operator given by a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
. More precisely, one of the results of
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 ...
is a
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 ...
that states that every
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
on a
Hilbert space is
unitarily equivalent to a multiplication operator on an
''L''''2'' space.
Example
Consider the
Hilbert space of
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued
square integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
functions on the
interval . With , define the operator
for any function in . This will be a
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a st ...
bounded linear 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 vect ...
, with domain all of and with
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
. Its
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 ...
will be the interval (the
range of the function defined on ). Indeed, for any complex number , the operator is given by
It is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
is not in , and then its inverse is
which is another multiplication operator.
This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any
''L''''p'' space.
See also
*
Translation operator
*
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 ...
*
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
...
*
Decomposition of spectrum (functional analysis)
Notes
References
*
{{DEFAULTSORT:Multiplication Operator
Functional analysis