Multiplication operator

Template:Distinguish In operator theory, a multiplication operator is an operator Template:Math defined on some vector space of functions and whose value at a function Template:Mvar is given by multiplication by a fixed function Template:Mvar. That is, $T_{f} φ (x) = f (x) φ (x)$ for all Template:Mvar in the domain of Template:Math, and all Template:Mvar in the domain of Template:Mvar (which is the same as the domain of Template:Mvar).^[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 Template:Mvar. 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 Template:Mvar is $σ$ -finite, is bounded if and only if Template:Mvar 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 Template:Mvar. As a consequence, $T_{f}$ is self-adjoint if and only if Template:Mvar is real-valued.^[4]

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

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

Consider the Hilbert space Template:Math of complex-valued square integrable functions on the interval Template:Closed-closed. With Template:Math, define the operator $T_{f} φ (x) = x^{2} φ (x)$ for any function Template:Mvar in Template:Mvar. This will be a self-adjoint bounded linear operator, with domain all of Template:Math and with norm Template:Math. Its spectrum will be the interval Template:Closed-closed (the range of the function Template:Math defined on Template:Closed-closed). Indeed, for any complex number Template:Mvar, the operator Template:Math is given by $(T_{f} - λ) (φ) (x) = (x^{2} - λ) φ (x) .$

It is invertible if and only if Template:Mvar is not in Template:Closed-closed, and then its inverse is $(T_{f} - λ)^{- 1} (φ) (x) = \frac{1}{x^{2} - λ} φ (x),$ 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.