the unit circle. For p = 1, one can perform a similar analysis. The results will not be exactly the same, since reflexivity no longer holds. Self-adjoint operators on Hilbert space Hilbert spaces are Banach spaces, so the above discussion applies to bounded operators on Hilbert spaces as well. A subtle point concerns the spectrum of T*. For a Banach space, T* denotes the transpose and σ(T*) = σ(T). For a Hilbert space, T* normally denotes the adjoint of an operator T ∈ B(H), not the transpose, and σ(T*) is not σ(T) but rather its image under complex conjugation. For a self-adjoint T ∈ B(H), the Borel functional calculus gives additional ways to break up the spectrum naturally. Borel functional calculus This subsection briefly sketches the development of this calculus. The idea is to first establish the continuous functional calculus, and then pass to measurable functions via the Riesz–Markov–Kakutani representation theorem. For the continuous functional calculus, the key ingredients are the following: If T is self-adjoint, then for any polynomial P, the operator norm satisfies The Stone–Weierstrass theorem, which implies that the family of polynomials (with complex coefficients), is dense in C(σ(T)), the continuous functions on σ(T). The family C(σ(T)) is a Banach algebra when endowed with the uniform norm. So the mapping is an isometric homomorphism from a dense subset of C(σ(T)) to B(H). Extending the mapping by continuity gives f(T) for f ∈ C(σ(T)): let Pn be polynomials such that Pn → f uniformly and define f(T) = lim Pn(T). This is the continuous functional calculus. For a fixed h ∈ H, we notice that is a positive linear functional on C(σ(T)). According to the Riesz–Markov–Kakutani representation theorem a unique measure μh on σ(T) exists such that This measure is sometimes called the spectral measure associated to h. The spectral measures can be used to extend the continuous functional calculus to bounded Borel functions. For a bounded function g that is Borel measurable, define, for a proposed g(T) Via the polarization identity, one can recover (since H is assumed to be complex) and therefore g(T) h for arbitrary h. In the present context, the spectral measures, combined with a result from measure theory, give a decomposition of σ(T). Decomposition into absolutely continuous, singular continuous, and pure point Let h ∈ H and μh be its corresponding spectral measure on σ(T) ⊂ R. According to a refinement of Lebesgue's