Multiplication operator explained

In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,$$T\_f\backslash varphi(x)\; =\; f(x)\; \backslash varphi\; (x)\; \backslash quad$$for all in the domain of, and all in the domain of (which is the same as the domain of).^[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 . 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 is \sigma -finite, is bounded if and only if is in L ^{infty(X)} . In this case, its operator norm is equal to \|f\| _{infty} .

• The adjoint of a multiplication operator

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

• The spectrum of a bounded multiplication operator

T _{f} is the essential range of ; outside of this spectrum, the inverse of (T _{f} - λ ) is the multiplication operator T _{(1)/(f -λ)} .

• Two bounded multiplication operators

T _{f} and T _{g} on L ^{2} 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$$T\_f\backslash varphi(x)\; =\; x^2\; \backslash varphi\; (x)$$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$$(T\_f\; -\; \backslash lambda)(\backslash varphi)(x)\; =\; (x^2-\backslash lambda)\; \backslash varphi(x).$$

It is invertible if and only if is not in, and then its inverse is $$(T\_f\; -\; \backslash lambda)^(\backslash varphi)(x)\; =\; \backslash frac\; \backslash varphi(x),$$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.

See also

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

References

• Book: Conway, J. B.. John B. Conway

. John B. Conway. A Course in Functional Analysis. 1990. Graduate Texts in Mathematics. 96. Springer Verlag. 0-387-97245-5.

Notes and References

1. Book: Arveson, William. William Arveson

   . William Arveson. A Short Course on Spectral Theory. 2001. Graduate Texts in Mathematics. 209. Springer Verlag. 0-387-95300-0.

2. Book: Halmos, Paul. Paul Halmos

   . Paul Halmos. A Hilbert Space Problem Book. Graduate Texts in Mathematics. 19. Springer Verlag. 1982. 0-387-90685-1.

3. Book: Weidmann, Joachim. Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics. 68. Springer Verlag. 1980. 978-1-4612-6029-5.
4. Book: Garcia. Stephan Ramon. Stephan Ramon Garcia. Mashreghi. Javad. Javad Mashreghi. Ross. William T.. Operator Theory by Example. 2023. Oxford Graduate Texts in Mathematics. 30. Oxford University Press. 9780192863867.