Definitions

# Multiplication operator

In operator theory, a multiplication operator is a linear 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,

$T\left(varphi\right)\left(x\right) = f\left(x\right) varphi \left(x\right) quad$

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 L2 space.

## Example

Consider the Hilbert space X=L2[−1, 3] of complex-valued square integrable functions on the interval [−1, 3]. Define the operator:

$T\left(varphi\right)\left(x\right) = x^2 varphi \left(x\right) quad$
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 xx2 defined on [−1, 3]). Indeed, for any complex number λ, the operator T-λ is given by
$\left(T-lambda\right)\left(varphi\right)\left(x\right) = \left(x^2-lambda\right) varphi\left(x\right). quad$
It is invertible if and only if λ is not in [0, 9], and then its inverse is

$\left(T-lambda\right)^\left\{-1\right\}\left(varphi\right)\left(x\right) = frac\left\{1\right\}\left\{x^2-lambda\right\} varphi\left(x\right) quad$

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

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