Motivation
If T is a self-adjoint operator on a finite dimensional inner product space H, H has an orthonormal basis
consisting of eigenvectors of T, that is
Thus, for any positive integer n,
In this case, given a Borel function h, we can define an operator h(T) by specifying its behavior on the basis:
In general, any self-adjoint operator T is unitarily equivalent to a multiplication operator; this means that for many purposes, T can be considered as an operator
acting on L2 of some measure space. The domain of T consists of those functions for which the above expression is in L2. In this case, we can define analogously
For many technical purposes, the preceding formulation is good enough. However, it is desirable to formulate the functional calculus in a way in which it is clear that it does not depend on the particular representation of T as a multiplication operator. This we do in the next section.
Read more about this topic: Borel Functional Calculus
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