Template:Distinguish In operator theory, a multiplication operator is a linear operator Template:Math defined on some vector space of functions and whose value at a function Template:Mvar is given by multiplication by a fixed function Template:Mvar. That is, <math display="block">T_f\varphi(x) = f(x) \varphi (x) \quad </math> for all Template:Mvar in the domain of Template:Math, and all Template:Mvar in the domain of Template:Mvar (which is the same as the domain of Template:Mvar).<ref name=arveson>Template:Cite book</ref>
Multiplication operators generalize the notion of operator given by a diagonal matrix.<ref>Template:Cite book</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 unitarily equivalent to a multiplication operator on an L2 space.<ref>Template:Cite book</ref>
These operators are often contrasted with composition operators, which are similarly induced by any fixed function Template:Mvar. They are also closely related to Toeplitz operators, which are compressions of multiplication operators on the circle to the Hardy space.
PropertiesEdit
- A multiplication operator <math>T_f</math> on <math>L^2(X)</math>, where Template:Mvar is <math>\sigma</math>-finite, is bounded if and only if Template:Mvar 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 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 Template:Mvar. As a consequence, <math>T_f</math> is self-adjoint if and only if Template:Mvar is real-valued.<ref name=garcia>Template:Cite book</ref>
- The spectrum of a bounded multiplication operator <math>T_f</math> is the essential range of Template:Mvar; 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 Template:Mvar and Template:Mvar are equal almost everywhere.<ref name=garcia></ref>
ExampleEdit
Consider the Hilbert space Template:Math of complex-valued square integrable functions on the interval Template:Closed-closed. With Template:Math, define the operator <math display="block">T_f\varphi(x) = x^2 \varphi (x) </math> for any function Template:Mvar in Template:Mvar. This will be a self-adjoint bounded linear operator, with domain all of Template:Math and with norm Template:Math. Its spectrum will be the interval Template:Closed-closed (the range of the function Template:Math defined on Template:Closed-closed). Indeed, for any complex number Template:Mvar, the operator Template:Math is given by <math display="block">(T_f - \lambda)(\varphi)(x) = (x^2-\lambda) \varphi(x). </math>
It is invertible if and only if Template:Mvar is not in Template:Closed-closed, 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.
See alsoEdit
- Translation operator
- Shift operator
- Transfer operator
- Decomposition of spectrum (functional analysis)