Example
Consider the Hilbert space X=L2 of complex-valued square integrable functions on the interval . Define the operator:
for any function φ in X. This will be a self-adjoint bounded linear operator 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-λ is given by
It is invertible if and only if λ is not in, and then its inverse is
which is another multiplication operator.
This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.
Read more about this topic: Multiplication Operator
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