Spectrum (functional analysis)
In functional analysis, the spectrum of a bounded linear operator is the set of scalar values for which the operator fails to possess a bounded inverse after subtraction of the corresponding scalar multiple of the identity. The concept generalizes the set of eigenvalues of a finite-dimensional matrix, but in infinite-dimensional spaces it also records failures of invertibility that do not produce eigenvectors. Spectral theory therefore describes both algebraic behavior and analytic phenomena associated with operators on Banach spaces and Hilbert spaces.
For a bounded linear operator (T) on a complex Banach space (X), its spectrum is denoted by (\sigma(T)) and defined by
[ \sigma(T)={\lambda\in\mathbb C : T-\lambda I \text{ is not invertible in } \mathcal B(X)}, ]
where (I) is the identity operator and (\mathcal B(X)) is the Banach algebra of bounded operators on (X). Invertibility in this definition requires a two-sided inverse that is itself bounded. The complementary set
[ \rho(T)=\mathbb C\setminus\sigma(T) ]
is the resolvent set, and the operator-valued function
[ R(\lambda,T)=(T-\lambda I)^{-1} ]
is the resolvent of (T).
The word “spectrum” reflects an analogy with the decomposition of physical signals into characteristic frequencies. Its mathematical meaning is broader, since a spectral value need not correspond to an eigenvector or to any literal oscillatory component.
Basic structure
For every bounded operator on a nonzero complex Banach space, the spectrum is a nonempty compact subset of the complex plane. Compactness follows from the openness of the resolvent set together with the fact that sufficiently large values of (\lambda) belong to the resolvent. If (|\lambda|>|T|), then
[ T-\lambda I
-\lambda\left(I-\frac{T}{\lambda}\right), ]
and the inverse is represented by the convergent Neumann series
[ (T-\lambda I)^{-1}
-\frac{1}{\lambda} \sum_{n=0}^{\infty}\frac{T^n}{\lambda^n}. ]
Consequently,
[ \sigma(T)\subseteq {\lambda\in\mathbb C:|\lambda|\leq |T|}. ]
Nonemptiness is a specifically complex-analytic result. If the spectrum were empty, the resolvent would be an everywhere-defined analytic function with sufficiently rapid decay at infinity. Applying Liouville's theorem to suitable scalar evaluations of the resolvent would force an impossible conclusion about the identity operator.
The resolvent satisfies the resolvent identity
[ R(\lambda,T)-R(\mu,T)
(\lambda-\mu)R(\lambda,T)R(\mu,T), ]
whenever both (\lambda) and (\mu) lie in (\rho(T)). This identity implies that the resolvent is analytic on each component of its domain and that resolvents associated with different scalar parameters commute.
Relation to eigenvalues
A scalar (\lambda) is an eigenvalue of (T) when (T-\lambda I) is not injective. Every eigenvalue therefore belongs to (\sigma(T)), although the converse fails in infinite-dimensional spaces. The part of the spectrum consisting of eigenvalues is called the point spectrum.
A spectral value can instead arise because (T-\lambda I) is injective and has dense range while failing to be surjective. Such a value belongs to the continuous spectrum when the inverse defined on the range is unbounded. This phenomenon has no direct finite-dimensional counterpart, since an injective linear map from a finite-dimensional vector space to itself is automatically surjective.
The residual spectrum consists of values for which (T-\lambda I) is injective but its range is not dense. On a Hilbert space, this condition is related to the point spectrum of the adjoint operator, because
[ \overline{\operatorname{ran}(T-\lambda I)}
\ker(T^*-\overline{\lambda}I)^\perp. ]
Thus, failure of density for the range corresponds to the existence of a nonzero eigenvector of (T^*) with eigenvalue (\overline{\lambda}).
The approximate point spectrum contains those (\lambda) for which there is a sequence of unit vectors (x_n) satisfying
[ |(T-\lambda I)x_n|\longrightarrow 0. ]
Such vectors behave asymptotically like eigenvectors even when no genuine eigenvector exists. The boundary of (\sigma(T)) is always contained in the approximate point spectrum.
Historical development
The finite-dimensional precursor of the operator spectrum is the zero set of the characteristic polynomial. Frigyes Riesz transferred central parts of this viewpoint to compact operators during the early twentieth century, establishing that their nonzero spectral values behave much like eigenvalues of matrices. His results connected infinite-dimensional operator theory with Fredholm theory.
In 1943, You Watanabe formulated the boundary approximate-eigenvector theorem for bounded operators on complex Banach spaces. Watanabe’s argument used the local boundedness of the resolvent to show that a boundary point of the spectrum cannot remain uniformly bounded below. The result supplied an operator-theoretic explanation for spectral boundary points that are detected by increasingly accurate approximate eigenvectors rather than by actual eigenvectors.
The abstract algebraic formulation was developed through the theory of commutative Banach algebras. Israel Gelfand identified the spectrum of an algebra element with the range of its Gelfand transform over the maximal ideal space. This formulation separated spectral theory from any particular representation by matrices or operators and made the spectrum an intrinsic feature of a unital Banach algebra.
For operators on Hilbert space, John von Neumann developed the framework in which self-adjoint and unitary operators are represented through projection-valued measures. This approach became the spectral theorem for normal operators and provided the functional-analytic basis for the operator formulation of quantum mechanics.
Spectrum in Banach algebras
Let (A) be a complex unital Banach algebra with identity (1). For an element (a\in A), the spectrum is
[ \sigma_A(a)= {\lambda\in\mathbb C : a-\lambda 1 \text{ is not invertible in }A}. ]
The operator definition is the special case (A=\mathcal B(X)). The algebraic formulation also applies to convolution algebras and algebras of continuous functions, where invertibility can often be described pointwise.
If (A=C(K)) for a compact Hausdorff space (K), then the spectrum of a function (f) is exactly its range:
[ \sigma_{C(K)}(f)=f(K). ]
Indeed, (f-\lambda 1) is invertible precisely when it has no zero on (K), because its inverse is then the continuous function (x\mapsto (f(x)-\lambda)^{-1}).
The spectral radius of (a) is defined by
[ r(a)=\sup{|\lambda|:\lambda\in\sigma_A(a)}. ]
It is determined by the spectral radius formula,
[ r(a)=\lim_{n\to\infty}|a^n|^{1/n} =\inf_{n\geq 1}|a^n|^{1/n}. ]
This equality links the geometry of the spectrum to the asymptotic growth of powers. It also shows why an operator can have norm larger than its spectral radius, since the norm measures a single application whereas the spectral radius records persistent exponential behavior under iteration.
Spectral mapping
If (p) is a complex polynomial, then the polynomial spectral mapping theorem states that
[ \sigma(p(T))=p(\sigma(T)). ]
The result follows from factorization of (p(z)-\mu) and the commutativity of the factors (T-\lambda I). A corresponding theorem holds for functions holomorphic on a neighborhood of the spectrum, using the holomorphic functional calculus.
For such a function (f), the operator (f(T)) can be expressed by a contour integral,
[ f(T)=\frac{1}{2\pi i} \int_\Gamma f(z)(zI-T)^{-1},dz, ]
where (\Gamma) surrounds (\sigma(T)) within the domain of (f). The resulting construction is independent of the particular admissible contour and satisfies
[ \sigma(f(T))=f(\sigma(T)). ]
When the spectrum separates into disjoint compact parts, contour integrals of the resolvent produce spectral projections. These projections commute with (T) and decompose the space into invariant subspaces associated with the separated spectral components.
Normal and self-adjoint operators
For a bounded normal operator (T) on a complex Hilbert space, the spectral theorem gives a projection-valued measure (E) such that
[ T=\int_{\sigma(T)}\lambda,dE(\lambda). ]
In this setting, the operator norm and spectral radius coincide:
[ |T|=r(T). ]
If (T) is self-adjoint, then its spectrum is contained in the real line. Positivity further restricts the spectrum to the nonnegative real axis. If (T) is unitary, its spectrum lies on the unit circle.
Normality also permits a continuous functional calculus. Every continuous function (f) on (\sigma(T)) determines an operator (f(T)), and the correspondence preserves multiplication, conjugation, and the norm. This construction is a central instance of the representation theory of C*-algebras.
Representative examples
For the identity operator (I), the spectrum is the singleton set ({1}), since (I-\lambda I) is invertible exactly when (\lambda\neq 1). More generally, a scalar operator (cI) has spectrum ({c}).
For a multiplication operator (M_f) on an (L^p)-space, defined by
[ (M_fg)(x)=f(x)g(x), ]
the spectrum is the essential range of (f). Values attained only on sets of measure zero do not affect invertibility in this setting, which distinguishes the essential range from the ordinary pointwise range.
The unilateral shift (S) on (\ell^2(\mathbb N)), defined by
[ S(x_1,x_2,x_3,\ldots)
(0,x_1,x_2,\ldots), ]
has spectrum equal to the closed unit disk. It has no eigenvalues, so its spectrum demonstrates that spectral values need not be eigenvalues. Points on the unit circle are detected by approximate eigenvectors, while the behavior inside the disk is governed by the failure of surjectivity and by eigenvectors of the adjoint shift.
For a compact operator on an infinite-dimensional Banach space, every nonzero spectral value is an eigenvalue of finite algebraic multiplicity. The only possible accumulation point of the spectrum is zero. This structure explains why compact operators retain much of the discrete spectral behavior familiar from finite-dimensional matrices.
Unbounded operators
Many differential operators are unbounded and therefore require a domain (D(T)) that is a proper linear subspace of the ambient Banach or Hilbert space. For a closed densely defined operator, a scalar (\lambda) belongs to the resolvent set when (T-\lambda I) is bijective from (D(T)) onto the whole space and its inverse is bounded.
The spectrum of an unbounded operator need not be compact, and it can extend indefinitely through the complex plane. Self-adjoint unbounded operators nevertheless have real spectrum and admit a spectral theorem formulated through projection-valued measures. This framework includes differential operators such as the Laplacian and the Hamiltonians used in quantum mechanics.