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 operator 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 that states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L² space.