Multiplication operator

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(varphi)(x) = f(x) varphi (x) 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.


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

T(varphi)(x) = x^2 varphi (x) 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
(T-lambda)(varphi)(x) = (x^2-lambda) varphi(x). quad
It is invertible if and only if λ is not in [0, 9], and then its inverse is

(T-lambda)^{-1}(varphi)(x) = frac{1}{x^2-lambda} varphi(x) 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.

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