# Multiplication operator

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In operator theory, a **multiplication operator** is an operator *T* 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 function space 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]. Define the operator:

for any function φ in *X*. This will be a self-adjoint bounded linear operator 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*-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.