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 $f$ ).^[1]

Multiplication operators generalize the notion of operator given by a diagonal matrix.^[2] 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.^[3]

These operators are often contrasted with composition operators, which are similarly induced by any fixed function $f$ . 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 $T_{f}$ on $L^{2}(X)$ , where $X$ is $\sigma$ -finite, is bounded if and only if $f$ is in $L^{\infty }(X)$ . In this case, its operator norm is equal to $\|f\|_{\infty }$ .^[1]

The adjoint of a multiplication operator $T_{f}$ is $T_{\overline {f}}$ , where ${\overline {f}}$ is the complex conjugate of $f$ . As a consequence, $T_{f}$ is self-adjoint if and only if $f$ is real-valued.^[4]

The spectrum of a bounded multiplication operator $T_{f}$ is the essential range of $f$ ; outside of this spectrum, the inverse of $(T_{f}-\lambda )$ is the multiplication operator $T_{\frac {1}{f-\lambda }}.$ ^[1]

Two bounded multiplication operators $T_{f}$ and $T_{g}$ on $L^{2}$ are equal if $f$ and $g$ are equal almost everywhere.^[4]

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 $T_{f}\varphi (x)=x^{2}\varphi (x)$ for any function $φ$ in $X$ . This will be a self-adjoint bounded linear operator, with domain all of $X = L 2 [-1, 3]$ and with norm $9$ . Its spectrum will be the interval $[0, 9]$ (the range of the function $x \mapsto x 2$ defined on $[-1, 3]$ ). Indeed, for any complex number $λ$ , the operator $T f - λ$ is given by $(T_{f}-\lambda )(\varphi )(x)=(x^{2}-\lambda )\varphi (x).$

It is invertible if and only if $λ$ is not in $[0, 9]$ , and then its inverse is $(T_{f}-\lambda )^{-1}(\varphi )(x)={\frac {1}{x^{2}-\lambda }}\varphi (x),$ 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.

References

^ ^a ^b ^c Arveson, William (2001). A Short Course on Spectral Theory. Graduate Texts in Mathematics. Vol. 209. Springer Verlag. ISBN 0-387-95300-0.
^ Halmos, Paul (1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19. Springer Verlag. ISBN 0-387-90685-1.
^ Weidmann, Joachim (1980). Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics. Vol. 68. Springer Verlag. ISBN 978-1-4612-6029-5.
^ ^a ^b Garcia, Stephan Ramon; Mashreghi, Javad; Ross, William T. (2023). Operator Theory by Example. Oxford Graduate Texts in Mathematics. Vol. 30. Oxford University Press. ISBN 9780192863867.

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

[arveson-1] Arveson, William (2001). A Short Course on Spectral Theory. Graduate Texts in Mathematics. Vol. 209. Springer Verlag. ISBN 0-387-95300-0.

[2] Halmos, Paul (1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19. Springer Verlag. ISBN 0-387-90685-1.

[3] Weidmann, Joachim (1980). Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics. Vol. 68. Springer Verlag. ISBN 978-1-4612-6029-5.

[garcia-4] Garcia, Stephan Ramon; Mashreghi, Javad; Ross, William T. (2023). Operator Theory by Example. Oxford Graduate Texts in Mathematics. Vol. 30. Oxford University Press. ISBN 9780192863867.

[1]

[2]

[3]

[4]

Properties

Example

See also

References