Editing Multiplication operator (section)

== 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]].