decomposition theorem, μh can be decomposed into three mutually singular parts: where μac is absolutely continuous with respect to the Lebesgue measure, μsc is singular with respect to the Lebesgue measure and atomless, and μpp is a pure point measure. All three types of measures are invariant under linear operations. Let Hac be the subspace consisting of vectors whose spectral measures are absolutely continuous with respect to the Lebesgue measure. Define Hpp and Hsc in analogous fashion. These subspaces are invariant under T. For example, if h ∈ Hac and k = T h. Let χ be the characteristic function of some Borel set in σ(T), then So and k ∈ Hac. Furthermore, applying the spectral theorem gives This leads to the following definitions: The spectrum of T restricted to Hac is called the absolutely continuous spectrum of T, σac(T). The spectrum of T restricted to Hsc is called its singular spectrum, σsc(T). The set of eigenvalues of T is called the pure point spectrum of T, σpp(T). The closure of the eigenvalues is the spectrum of T restricted to Hpp. So Comparison A bounded self-adjoint operator on Hilbert space is, a fortiori, a bounded operator on a Banach space. Therefore, one can also apply to T the decomposition of the spectrum that was achieved above for bounded operators on a Banach space. Unlike the Banach space formulation, the union need not be disjoint. It is disjoint when the operator T is of uniform multiplicity, say m, i.e. if T is unitarily equivalent to multiplication by λ on the direct sum for some Borel measures . When more than one measure appears in the above expression, we see that it is possible for the union of the three types of spectra to not be disjoint. If , λ is sometimes called an eigenvalue embedded in the absolutely continuous spectrum. When T is unitarily equivalent to multiplication by λ on the decomposition of σ(T) from Borel functional calculus is a refinement of the Banach space case. Physics The preceding comments can be extended to the unbounded self-adjoint operators since Riesz-Markov holds for locally compact Hausdorff spaces. In quantum mechanics, observables are self-adjoint operators, often not bounded, and their spectra are the possible outcomes of measurements. Absolutely continuous spectrum of a physical observable corresponds to free states of a system, while the pure point spectrum corresponds to bound states. The singular spectrum