Editing Multiplication operator (section)

== 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>