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,

$T_{f}\varphi (x)=f(x)\varphi (x)\quad$

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 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 X = L²[−1, 3] of complex-valued square integrable functions on the interval [−1, 3]. With f(x) = x², define the operator

$T_{f}\varphi (x)=x^{2}\varphi (x)\quad$

for any function φ in X. This will be a self-adjoint bounded linear operator, with domain all of X = L²[−1, 3] 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

$(T_{f}-\lambda )(\varphi )(x)=(x^{2}-\lambda )\varphi (x).\quad$

It is invertible if and only if λ is not in [0, 9], and then its inverse is

$(T_{f}-\lambda )^{-1}(\varphi )(x)={\frac {1}{x^{2}-\lambda }}\varphi (x)\quad$

which is another multiplication operator.

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

See also

• Translation operator
• shift 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.CS1 maint: ref=harv (link)