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 $f$).

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 $f$. 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 $$\|f\|_\infty$$.


 * 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.


 * 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 $$T_{\frac{1}{f - \lambda}}.$$


 * Two bounded multiplication operators $$T_f$$ and $$T_g$$ on $$L^2$$ are equal if $f$ and $g$ are equal almost everywhere.

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 $x↦ x^{2}$ 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 Lp space.