Editing Multiplication operator

{{distinguish|Multiplication sign}}
In [[operator theory]], a '''multiplication operator''' is a [[linear operator]] {{math|''T''<sub>''f''</sub>}} defined on some [[function space|vector space of functions]] and whose value at a function {{mvar|φ}} is given by multiplication by a fixed function {{mvar|f}}. That is,
<math display="block">T_f\varphi(x) = f(x) \varphi (x) \quad </math>
for all {{mvar|φ}} in the [[domain of a function|domain]] of {{math|''T''<sub>''f''</sub>}}, and all {{mvar|x}} in the domain of {{mvar|φ}} (which is the same as the domain of {{mvar|f}}).<ref name=arveson>{{cite book|last=Arveson|first=William|authorlink = William Arveson|title=A Short Course on Spectral Theory|year=2001|series=Graduate Texts in Mathematics|volume=209|publisher=[[Springer Verlag]]|isbn=0-387-95300-0}}</ref>

Multiplication operators generalize the notion of operator given by a [[diagonal matrix]].<ref>{{cite book|last=Halmos|first=Paul|authorlink=Paul Halmos|title=A Hilbert Space Problem Book|series=Graduate Texts in Mathematics|volume=19|publisher=[[Springer Verlag]]|year=1982|isbn=0-387-90685-1}}</ref> More precisely, one of the results of [[operator theory]] is a [[spectral theorem]] that states that every [[self-adjoint operator]] on a [[Hilbert space]] is [[self-adjoint operator|unitarily equivalent]] to a multiplication operator on an [[Lp space|''L''<sup>''2''</sup> space]].<ref>{{cite book|last=Weidmann|first=Joachim|title=Linear Operators in Hilbert Spaces|series=Graduate Texts in Mathematics|volume=68|publisher=[[Springer Verlag]]|year=1980|isbn=978-1-4612-6029-5}}</ref>

These operators are often contrasted with [[composition operator]]s, which are similarly induced by any fixed function {{mvar|f}}. They are also closely related to [[Toeplitz operator|Toeplitz operators]], which are [[dilation (operator theory)|compression]]s of multiplication operators on the circle to the [[Hardy space]].

== Properties ==
* A multiplication operator <math>T_f</math> on <math>L^2(X)</math>, where {{mvar|X}} is [[sigma-finite|<math>\sigma</math>-finite]], is [[bounded operator|bounded]] if and only if {{mvar|f}} is in <math>L^\infty(X)</math>. (The backward direction of the implication does not require the <math>\sigma</math>-finiteness assumption.) In this case, its [[operator norm]] is equal to <math>\|f\|_\infty</math>.<ref name=arveson></ref>
* The [[Hermitian adjoint|adjoint]] of a multiplication operator <math>T_f</math> is <math>T_\overline{f}</math>, where <math>\overline{f}</math> is the [[complex conjugate]] of {{mvar|f}}. As a consequence, <math>T_f</math> is self-adjoint if and only if {{mvar|f}} is real-valued.<ref name=garcia>{{cite book|last1=Garcia|first1=Stephan Ramon|author1link = Stephan Ramon Garcia|last2=Mashreghi|first2=Javad|author2link = Javad Mashreghi|last3=Ross|first3=William T.|title=Operator Theory by Example|year=2023|series=Oxford Graduate Texts in Mathematics|volume=30|publisher=[[Oxford University Press]]|isbn=9780192863867}}</ref>
* The [[spectrum (functional analysis)|spectrum]] of a bounded multiplication operator <math>T_f</math> is the [[essential range]] of {{mvar|f}}; outside of this spectrum, the inverse of <math>(T_f - \lambda)</math> is the multiplication operator <math>T_{\frac{1}{f - \lambda}}.</math><ref name=arveson></ref>
* Two bounded multiplication operators <math>T_f</math> and <math>T_g</math> on <math>L^2</math> are equal if {{mvar|f}} and {{mvar|g}} are equal [[almost everywhere]].<ref name=garcia></ref>

== Example ==
Consider the [[Hilbert space]] {{math|1=''X'' = ''L''<sup>2</sup>[−1, 3]}} of [[complex number|complex]]-valued [[square integrable]] functions on the [[interval (mathematics)|interval]] {{closed-closed|−1, 3}}. With {{math|1=''f''(''x'') = ''x''<sup>2</sup>}}, define the operator
<math display="block">T_f\varphi(x) = x^2 \varphi (x) </math>
for any function {{mvar|φ}} in {{mvar|X}}. This will be a [[self-adjoint operator|self-adjoint]] [[bounded linear operator]], with domain all of {{math|1=''X'' = ''L''<sup>2</sup>[−1, 3]}} and with [[operator norm|norm]] {{math|9}}. Its [[spectrum of an operator|spectrum]] will be the interval {{closed-closed|0, 9}} (the [[range of a function|range]] of the function {{math|''x''↦ ''x''<sup>2</sup>}} defined on {{closed-closed|−1, 3}}). Indeed, for any complex number {{mvar|λ}}, the operator {{math|''T''<sub>''f''</sub> − ''λ''}} is given by
<math display="block">(T_f - \lambda)(\varphi)(x) = (x^2-\lambda) \varphi(x). </math>

It is [[invertible function|invertible]] [[if and only if]] {{mvar|λ}} is not in {{closed-closed|0, 9}}, and then its inverse is 
<math display="block">(T_f - \lambda)^{-1}(\varphi)(x) = \frac{1}{x^2-\lambda} \varphi(x),</math>
which is another multiplication operator.

This example can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any [[Lp space|''L''<sup>''p''</sup> space]].

== See also ==
* [[Translation operator (disambiguation)|Translation operator]]
* [[Shift operator]]
* [[Transfer operator]]
* [[Decomposition of spectrum (functional analysis)]]

== References ==
{{reflist}}

==Bibliography==
*{{cite book|last=Conway|first=J. B.|authorlink = John B. Conway|title=A Course in Functional Analysis|year=1990|series=Graduate Texts in Mathematics|volume=96|publisher=[[Springer Verlag]]|isbn=0-387-97245-5}}
{{DEFAULTSORT:Multiplication Operator}}
[[Category:Operator theory]]
[[Category:Linear operators]]