Multiplication operator

Template:Short description In operator theory, a multiplication operator is a linear operator T_f 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, $$T_f\phi (x)=f(x)\phi (x)$$ for all Template:Mvar in the domain of T_f, and all Template:Mvar in the domain of Template:Mvar (which is the same as the domain of Template:Mvar).

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 that states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L² space.

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 $\sigma$-finite, is bounded if and only if Template:Mvar is in $L^{\mathrm{\infty }}(X)$. (The backward direction of the implication does not require the $\sigma$-finiteness assumption.) In this case, its operator norm is equal to $\Vert f{\Vert }_{\mathrm{\infty }}$.
• The adjoint of a multiplication operator $T_f$ is $T_{\bar{f}}$, where $\bar{f}$ is the complex conjugate of Template:Mvar. As a consequence, $T_f$ is self-adjoint if and only if Template:Mvar is real-valued.
• The spectrum of a bounded multiplication operator $T_f$ is the essential range of Template:Mvar; outside of this spectrum, the inverse of $(T_f-\lambda )$ is the multiplication operator $T_{\frac{1}{f-\lambda }}.$
• Two bounded multiplication operators $T_f$ and $T_g$ on $L^2$ are equal if Template:Mvar and Template:Mvar are equal almost everywhere.

Example

Consider the Hilbert space X = L²[−1, 3] of complex-valued square integrable functions on the interval Template:Closed-closed. With f(x) = x², define the operator $$T_f\phi (x)=x^2\phi (x)$$ for any function Template:Mvar in Template:Mvar. This will be a self-adjoint bounded linear operator, with domain all of X = L²[−1, 3] and with norm 9. Its spectrum will be the interval Template:Closed-closed (the range of the function x↦ x² defined on Template:Closed-closed). Indeed, for any complex number Template:Mvar, the operator T_f − λ is given by $$(T_f-\lambda )(\phi )(x)=(x^2-\lambda )\phi (x).$$

It is invertible if and only if Template:Mvar is not in Template:Closed-closed, and then its inverse is $$(T_f-\lambda {)}^{-1}(\phi )(x)=\frac{1}{x^2-\lambda }\phi (x),$$ 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)