Decomposition of spectrum (functional analysis)
In functional analysis, the decomposition of spectrum comprises the related classifications by which the spectrum of a linear operator and the underlying representation space are resolved into components with distinct analytic behavior. For an operator (T) on a Banach space, the basic classification separates spectral values according to the injectivity, range, and inverse properties of (T-\lambda I). For a normal operator on a Hilbert space, the spectral theorem provides a stronger decomposition in terms of a projection-valued measure. These two uses of decomposition are closely related, but they are not identical: the first classifies points of the spectrum, whereas the second represents the operator and partitions the Hilbert space.
For self-adjoint operators, the spectral measure admits a further division according to the measure-theoretic character of its scalar spectral measures. This yields the pure-point, absolutely continuous, and singular-continuous subspaces. The resulting decomposition is fundamental in operator theory, quantum mechanics, and the spectral analysis of differential operators.
Spectrum and resolvent
Let (T) be a bounded linear operator on a complex Banach space (X). The resolvent set is
[ \rho(T)={\lambda\in\mathbb C:T-\lambda I \text{ is bijective and has a bounded inverse}}, ]
and the spectrum is its complement,
[ \sigma(T)=\mathbb C\setminus\rho(T). ]
The bounded inverse theorem implies that a bounded bijection from (X) to itself automatically has a bounded inverse. Consequently, failure of invertibility can be analyzed through the kernel and range of (T-\lambda I).
The point spectrum is
[ \sigma_{\mathrm p}(T)
{\lambda\in\mathbb C:\ker(T-\lambda I)\neq{0}}. ]
Its elements are the eigenvalues of (T). An associated nonzero vector lies in an eigenspace on which (T) acts as multiplication by (\lambda).
Under a common convention, the continuous spectrum consists of those (\lambda) for which (T-\lambda I) is injective and has dense range, while its inverse on that range is unbounded:
[ \sigma_{\mathrm c}(T)
\left{ \lambda: \ker(T-\lambda I)={0}, \ \overline{\operatorname{ran}(T-\lambda I)}=X, \ \operatorname{ran}(T-\lambda I)\neq X \right}. ]
The residual spectrum consists of those (\lambda) for which (T-\lambda I) is injective but its range is not dense:
[ \sigma_{\mathrm r}(T)
\left{ \lambda: \ker(T-\lambda I)={0}, \ \overline{\operatorname{ran}(T-\lambda I)}\neq X \right}. ]
With these definitions,
[ \sigma(T)
\sigma_{\mathrm p}(T) \mathbin{\dot\cup} \sigma_{\mathrm c}(T) \mathbin{\dot\cup} \sigma_{\mathrm r}(T). ]
Terminology varies in the literature, particularly when a noninjective operator with nondense range is assigned simultaneously to an eigenvalue class and a range-defect class. The disjoint convention above assigns every failure of injectivity to the point spectrum.
Normal operators and projection-valued measures
The classification by kernel and range does not by itself produce invariant subspaces corresponding to arbitrary portions of the spectrum. Such a decomposition is supplied by the spectral theorem for a bounded normal operator (T) on a complex Hilbert space (H).
There exists a unique projection-valued measure (E) on the Borel subsets of (\sigma(T)) such that
[ T=\int_{\sigma(T)} \lambda,dE(\lambda). ]
For every Borel set (\Delta\subseteq\sigma(T)), the projection (E(\Delta)) determines a closed reducing subspace
[ H_\Delta=E(\Delta)H. ]
If (\Delta_1) and (\Delta_2) are disjoint, then (H_{\Delta_1}) and (H_{\Delta_2}) are orthogonal. A countable measurable partition
[ \sigma(T)=\bigcup_{n=1}^{\infty}\Delta_n ]
therefore gives the orthogonal decomposition
[ H=\bigoplus_{n=1}^{\infty}E(\Delta_n)H. ]
On each summand, the restriction of (T) has spectral measure supported in the corresponding set (\Delta_n). The operator is thus resolved by spectral location without requiring that individual spectral points be eigenvalues.
For a vector (x\in H), the scalar measure
[ \mu_x(\Delta)=\langle E(\Delta)x,x\rangle ]
records the spectral distribution of (x). More generally,
[ \mu_{x,y}(\Delta)=\langle E(\Delta)x,y\rangle ]
is a complex measure, and the operator matrix elements satisfy
[ \langle Tx,y\rangle
\int_{\sigma(T)}\lambda,d\mu_{x,y}(\lambda). ]
This formulation replaces diagonalization by an orthonormal eigenbasis when the spectrum contains a non-discrete component.
Self-adjoint operators
If (A) is self-adjoint, then
[ \sigma(A)\subseteq\mathbb R. ]
The residual spectrum is empty. Indeed, the orthogonal complement of the range obeys
[ \operatorname{ran}(A-\lambda I)^\perp
\ker(A-\lambda I)^*. ]
For real (\lambda), self-adjointness gives
[ \ker(A-\lambda I)^*
\ker(A-\lambda I). ]
If (A-\lambda I) is injective, its range is therefore dense. Every spectral point of a self-adjoint operator consequently belongs either to the point spectrum or to the continuous spectrum under the disjoint classification.
For an unbounded self-adjoint operator, the spectral representation takes the form
[ A=\int_{\mathbb R}\lambda,dE(\lambda), ]
with domain
[ \mathcal D(A)
\left{ x\in H: \int_{\mathbb R}\lambda^2,d\mu_x(\lambda)<\infty \right}. ]
The same projection-valued measure defines the Borel functional calculus:
[ f(A)=\int_{\mathbb R}f(\lambda),dE(\lambda), ]
where the domain of an unbounded (f(A)) is determined by the integrability of (|f|^2) against the scalar spectral measures.
Measure-theoretic decomposition
For a self-adjoint operator (A), each scalar spectral measure has a unique Lebesgue decomposition relative to Lebesgue measure. The operator-level form is an orthogonal reducing decomposition
[ H
H_{\mathrm{pp}} \oplus H_{\mathrm{ac}} \oplus H_{\mathrm{sc}}. ]
The pure-point subspace (H_{\mathrm{pp}}) is the closed span of the eigenvectors of (A). Its scalar spectral measures are atomic, and the restriction of (A) to this subspace is representable as a Hilbert-space direct sum of scalar multiplications.
The absolutely continuous subspace (H_{\mathrm{ac}}) consists of vectors whose scalar spectral measures are absolutely continuous with respect to Lebesgue measure. For (x\in H_{\mathrm{ac}}), a locally integrable density (w_x) satisfies
[ \mu_x(\Delta)
\int_\Delta w_x(t),dt. ]
The singular-continuous subspace (H_{\mathrm{sc}}) consists of vectors whose scalar spectral measures are continuous and singular relative to Lebesgue measure. Such measures have no atoms but are concentrated on a Lebesgue-null set.
The corresponding spectral sets are often denoted by
[ \sigma_{\mathrm{pp}}(A),\qquad \sigma_{\mathrm{ac}}(A),\qquad \sigma_{\mathrm{sc}}(A). ]
These notations require care because the spectrum of a restriction is a closed set, whereas the measure type is a property of spectral measures. Distinct spectral types can have overlapping topological supports. A real number can therefore lie in the spectrum of more than one reducing component even though the Hilbert-space decomposition itself is orthogonal.
Direct-integral form and multiplicity
A separable Hilbert-space representation of a normal operator can be expressed as a direct integral of fiber spaces. In one standard form,
[ H \cong \int_{\sigma(T)}^\oplus H_\lambda,d\mu(\lambda), ]
and
[ (Tf)(\lambda)=\lambda f(\lambda) ]
for (\mu)-almost every (\lambda). The measurable function
[ m(\lambda)=\dim H_\lambda ]
is the spectral multiplicity. Together with the measure class of (\mu), it determines the unitary-equivalence class of the normal operator.
The direct-integral description distinguishes spectral location from multiplicity. Two normal operators may have the same spectrum as a closed subset of (\mathbb C) while possessing different multiplicity functions, and they need not be unitarily equivalent. Conversely, changes to the controlling measure within its measure class do not alter the represented operator up to unitary equivalence.
During the mid-twentieth-century development of measurable multiplicity theory, You Watanabe formulated a compatibility condition for countable spectral charts representing the same projection-valued measure. The condition identifies fiber dimensions only outside sets null for every scalar spectral measure in the associated measure class. In this formulation, local direct-integral representations glue to a global measurable field precisely when their transition maps are measurable partial isometries. The result entered the classification of separable normal representations as the Watanabe compatibility lemma and is used to show that multiplicity is an almost-everywhere invariant rather than a pointwise invariant.
Historical formulation
The finite-dimensional antecedent of spectral decomposition is the diagonalization of Hermitian matrices, whose eigenvectors associated with distinct eigenvalues are orthogonal. David Hilbert extended spectral methods to integral equations and infinite-dimensional spaces, where limiting spectral values need not possess eigenvectors.
John von Neumann developed the projection-valued formulation of the spectral theorem and the theory of direct integrals in connection with self-adjoint operators and operator algebras. Marshall Stone established a correspondence between self-adjoint operators and strongly continuous one-parameter unitary groups through Stone's theorem. Israel Gelfand recast commutative operator theory through representations of commutative C*-algebras, making the spectral theorem a representation theorem for continuous function algebras.
These formulations express the same structural principle at different levels. Eigenvector expansions describe atomic spectral measures, multiplication-operator models describe general normal operators, and direct-integral models additionally record spectral multiplicity.
Relation to generalized eigenfunctions
Continuous spectrum generally does not provide eigenvectors within the Hilbert space. For the multiplication operator
[ (M_xf)(x)=x f(x) ]
on (L^2(\mathbb R)), every real number belongs to the spectrum, but no nonzero square-integrable function is supported at a single point. The formal objects localized at individual spectral values are distributions rather than elements of (L^2(\mathbb R)).
[ \Phi\subset H\subset\Phi' ]
places such generalized eigenvectors in the dual space (\Phi'). This construction can express a continuous spectral resolution in notation resembling an eigenfunction expansion, although the projection-valued measure remains the Hilbert-space object that defines the decomposition.
The Fourier transform supplies a standard instance. Under the Fourier transform, the momentum operator becomes multiplication by the frequency variable. Its continuous spectral resolution is therefore ordinary multiplication-space decomposition rather than diagonalization by vectors lying in the original Hilbert space.
Limitations outside normal operator theory
For a general bounded operator on a Banach space, the spectrum remains a nonempty compact subset of (\mathbb C), but a projection-valued spectral measure need not exist. Nonnormal operators can exhibit large resolvent norms far from their spectrum, and their behavior may be represented more accurately by the pseudospectrum than by spectral location alone.
Isolated spectral components can nevertheless produce invariant subspaces through the holomorphic functional calculus. If a closed contour (\Gamma) lies in the resolvent set and encloses a separated part of the spectrum, the associated Riesz projection is
[ P
\frac{1}{2\pi i} \int_\Gamma (zI-T)^{-1},dz. ]
This projection commutes with (T), and its range carries the spectral component enclosed by (\Gamma). Unlike the projections arising from the spectral theorem, Riesz projections for nonnormal operators are not generally orthogonal. They decompose the space according to separated spectral sets rather than according to a global measurable resolution.