In operator theory, a multiplication operator is an operator Tf 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 Tf, 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]. With f(x) = x2, define the operator
for any function φ in X. This will be a self-adjoint bounded linear operator, with domain all of X = L2[−1, 3] with norm 9. Its spectrum will be the interval [0, 9] (the range of the function x→ x2 defined on [−1, 3]). Indeed, for any complex number λ, the operator Tf − 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.