The General Functional Calculus
We can also define the functional calculus for not necessarily bounded Borel functions h; the result is an operator which in general fails to be bounded. Using the multiplication by a function f model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition of h with f.
Theorem. Let T be a self-adjoint operator on H, h a real-valued Borel function on R. There is a unique operator S such that
The operator S of the previous theorem is denoted h(T).
More generally, a Borel functional calculus also exists for (bounded) normal operators.
Read more about this topic: Borel Functional Calculus
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