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,
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:
for any function φ in X
. This will be a self-adjoint bounded linear operator
9. Its spectrum
will be the interval [0, 9] (the range
of the function x
defined on [−1, 3]). Indeed, for any complex number λ, the operator T
-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.