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 $X = L 2 [-1, 3]$ of complex-valued square integrable functions on the interval $[-1, 3]$ . With $f (x) = x 2$ , 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 $X = L 2 [-1, 3]$ with norm $9$ . Its spectrum will be the interval $[0, 9]$ (the range of the function $x \to x 2$ 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.

Notes

References

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

Multiplication operator

Example

See also

Notes

References