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 for all in the domain of , and all in the domain of (which is the same as 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 compressionss of multiplication operators on the circle to the Hardy space.

Properties

• , where is \sigma-finite, is bounded if and only if -finiteness assumption.) In this case, its operator norm.

• The adjoint is the complex conjugate of is self-adjoint if and only if is real-valued.

• The spectrum is the essential range of

• are equal if and are equal almost everywhere.

Example

Consider the Hilbert space of complex-valued square integrable functions on the interval . With 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

It is invertible if and only if is not in which is another multiplication operator.

This example can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.

See also

• Shift operator

• Transfer operator

• Decomposition of spectrum (functional analysis)

References

Bibliography