Multiplication operator

In operator theory, a multiplication operator is an operator T_f 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, [math]\displaystyle{ T_f\varphi(x) = f(x) \varphi (x) \quad }[/math] for all φ in the domain of T_f, and all x in the domain of φ (which is the same as the domain of f).

This type of operator 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 that 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 X = L²[−1, 3] of complex-valued square integrable functions on the interval [−1, 3]. With f(x) = x², define the operator [math]\displaystyle{ T_f\varphi(x) = x^2 \varphi (x) }[/math] for any function φ in X. 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]). Indeed, for any complex number λ, the operator T_f − λ is given by [math]\displaystyle{ (T_f - \lambda)(\varphi)(x) = (x^2-\lambda) \varphi(x). }[/math]

It is invertible if and only if λ is not in [0, 9], and then its inverse is [math]\displaystyle{ (T_f - \lambda)^{-1}(\varphi)(x) = \frac{1}{x^2-\lambda} \varphi(x), }[/math] 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

• Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. 96. Springer Verlag. ISBN 0-387-97245-5.