Multiplication operator
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 .
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.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 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.
