Multiplication operator

In operator theory, a multiplication operator is a linear 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 .
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.
These operators are often contrasted with composition operators, which are similarly induced by any fixed function. They are also closely related to Toeplitz operators, which are compressions of multiplication operators on the circle to the Hardy space.

Properties

A multiplication operator on, where is -finite, is bounded if and only if is in. In this case, its operator norm is equal to.
The adjoint of a multiplication operator is, where is the complex conjugate of. As a consequence, is self-adjoint if and only if is real-valued.
The spectrum of a bounded multiplication operator is the essential range of ; outside of this spectrum, the inverse of is the multiplication operator
Two bounded multiplication operators and on are equal if and are equal almost everywhere.

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 . 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 example can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any L^p space.