In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is, for all in the domain of , and all in the domain of (which is the same as the domain of ). 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. Example Consider the Hilbert space of complex-valued square integrable functions on the interval . With , define the operator for any function in . This will be a self-adjoint bounded linear operator, with domain all of and with norm . Its spectrum will be the interval (the range of the function defined on ). Indeed, for any complex number , the operator is given by It is invertible if and only if is not in , 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. See also Translation operator Shift operator Transfer operator Decomposition of spectrum (functional analysis) Notes References Functional analysis