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 (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 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 . (The backward direction of the implication does not require the -finiteness assumption.) 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.

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

See also

• Translation operator
• Shift operator
• Transfer operator
• Decomposition of spectrum (functional analysis)

References

Bibliography