# Multiplication operator

In operator theory, a **multiplication operator** is an operator *T*_{f} defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is,

for all φ in the domain of *T*_{f}, and all x in the domain of φ (which is the same as the domain of f).

This type of operators is often contrasted with composition operators. 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, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an *L*^{2} space.

## Example

Consider the Hilbert space *X* = *L*^{2}[−1, 3] of complex-valued square integrable functions on the interval [−1, 3]. With *f*(*x*) = *x*^{2}, define the operator

for any function φ in X. This will be a self-adjoint bounded linear operator, with domain all of *X* = *L*^{2}[−1, 3] with norm 9. Its spectrum will be the interval [0, 9] (the range of the function *x*→ *x*^{2} defined on [−1, 3]). Indeed, for any complex number λ, the operator *T*_{f} − *λ* is given by

It is invertible if and only if λ is not in [0, 9], and then its inverse is

which is another multiplication operator.

This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.

## See also

## Notes

## References

- Conway, J. B. (1990).
*A Course in Functional Analysis*. Graduate Texts in Mathematics.**96**. Springer Verlag. ISBN 0-387-97245-5.