## Abstract

The apparatus of functions of an operator was created to enable the free use of analogies between formulas involving numbers and formulas that involve operators. In this chapter we build this apparatus starting with polynomials in an operator, then extending the definition to continuous functions and finally to bounded Borel-measurable function of a self-adjoint operator. After giving a brief introduction into integration with respect to a vector measure, we define the spectral measure of a self-adjoint operator and demonstrate the representation of functions of an operator as integrals with respect to the spectral measure. Some applications are given.

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