multiplication operator

In operator theory, a multiplication operator is a linear operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
$T_f\varphi(x) = f(x) \varphi (x) \quad$
for all in the domain of , and all in the domain of (which is the same as the domain of ).

Multiplication operators generalize the notion of operator given by a diagonal matrix. 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 ''L''^''2'' space.

These operators are often contrasted with composition operators, which are similarly induced by any fixed function . They are also closely related to Toeplitz operators, which are compressions of multiplication operators on the circle to the Hardy space.

Properties

* A multiplication operator $T_f$ on $L^2(X)$, where is $\sigma$-finite, is bounded if and only if is in $L^\infty(X)$. (The backward direction of the implication does not require the $\sigma$-finiteness assumption.) In this case, its operator norm is equal to $\, f\, _\infty$.
* The adjoint of a multiplication operator $T_f$ is $T_\overline$, where $\overline$ is the complex conjugate of . As a consequence, $T_f$ is self-adjoint if and only if is real-valued.
* The spectrum of a bounded multiplication operator $T_f$ is the essential range of ; outside of this spectrum, the inverse of $(T_f - \lambda)$ is the multiplication operator $T_.$
* Two bounded multiplication operators $T_f$ and $T_g$ on $L^2$ are equal if and are equal almost everywhere.

Example

Consider the Hilbert space of complex-valued square integrable functions on the interval . With , define the operator
$T_f\varphi(x) = x^2 \varphi (x)$
for any function in . This will be a self-adjoint bounded linear operator, with domain all of and with norm . Its spectrum will be the interval (the range of the function defined on ). Indeed, for any complex number , the operator is given by
$(T_f - \lambda)(\varphi)(x) = (x^2-\lambda) \varphi(x).$

It is invertible if and only if is not in , and then its inverse is
$(T_f - \lambda)^(\varphi)(x) = \frac \varphi(x),$
which is another multiplication operator.

This example can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any ''L''^''p'' space.

See also

* Translation operator
* Shift operator
* Transfer operator
* Decomposition of spectrum (functional analysis)

References

Bibliography

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{{DEFAULTSORT:Multiplication Operator
Operator theory
Linear operators