What is called functional calculus or function calculus are operations by which for a function (on the complex numbers , for instance) and a suitable linear operator (on a Hilbert space, for instance) one makes sense of the expression as a new operator. Usually one requires that the assignment is an algebra homomorphism, but not always, namely in some contexts as quantization the ordering effects may not respect the homomorphism property, see Weyl functional calculus.
Let be a C-star algebra (possibly non-commutative) and a normal operator. With the operator spectrum of write for the commutative -algebra of contrinuous complex-valued functions on . Finally write for the function .
such that .
For all we have that is a normal operator.
This appears for instance as (KadisonRingrose, theorem 4.4.5).
Let be the -algebra generated by , or in fact any commutative -subalgebra of containing .
Define a morphism
by . This is a continuous -algebra homomorphism. Therefore so is the composite
And this satisfies .
This establishes the existence of . To see uniqueness, notice that any other morphism with these properties coincides with on all polynomials in and . By the Stone-Weierstrass theorem such polynomials form an everywhere-dense subset of . Since moreover one can see that the two morphisms must be isometric (…) it follows that they in fact agree.
A standard textbook is for instance