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,
[2] 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.
[3] These operators are often contrasted with composition operators, which are similarly induced by any fixed function f. They are also closely related to Toeplitz operators, which are compressions of multiplication operators on the circle to the Hardy space.
Consider the Hilbert space X = L2[−1, 3] of complex-valued square integrable functions on the interval [−1, 3].
This will be a self-adjoint bounded linear operator, with domain all of X = L2[−1, 3] and with norm 9.
Its spectrum will be the interval [0, 9] (the range of the function x↦ x2 defined on [−1, 3]).
It is invertible if and only if λ is not in [0, 9], and then its inverse is
This example can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.