Editing Multiplication operator (section)

{{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]].