Multiplication operator

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). (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 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 of complex-valued square integrable functions on the interval . With , 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 and with norm 9. Its spectrum will be the interval (the range of the function x↦ x2 defined on ). 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 , 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 L**p space.

References

Bibliography

References

1. Arveson, William. (2001). "A Short Course on Spectral Theory". [[Springer Verlag]].
2. Halmos, Paul. (1982). "A Hilbert Space Problem Book". [[Springer Verlag]].
3. Weidmann, Joachim. (1980). "Linear Operators in Hilbert Spaces". [[Springer Verlag]].
4. (2023). "Operator Theory by Example". [[Oxford University Press]].