Multiplication operator

• A multiplication operator T_f on L^2(X), where is \sigma-finite, is bounded if and only if is in L^\infty(X). (The backward direction of the implication does not require the \sigma-finiteness assumption.) 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. • The spectrum of a bounded multiplication operator T_f is the essential range of ; 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 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\varphi(x) = x^2 \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 - \lambda)(\varphi)(x) = (x^2-\lambda) \varphi(x). It is invertible if and only if is not in , 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. == See also ==