# Spectrum (functional analysis)

In functional analysis, the **spectrum** of an element a of a complex Banach algebra with unit is the set σ(a) of complex numbers λ for which a − λ1 is not invertible; for a bounded linear operator T on a [Banach space](https://www.edgepedia.org/banach-space), it is the set of λ for which T − λI has no bounded inverse. The spectrum generalizes the set of eigenvalues of a matrix, and it is the central object of spectral theory.

| Key fact | Detail |
|---|---|
| Definition | σ(a) = {λ ∈ C : a − λ1 is not invertible}; the resolvent set ρ(a) is its complement<sup>[1](https://www.math.ucsd.edu/~bdriver/241B_W2020/Lecture%20Notes/241Functional_2020_Ver3.pdf)</sup> |
| Compactness and nonemptiness | For a unital Banach algebra, σ(a) is compact, nonempty, and contained in {λ : |λ| ≤ ‖a‖}<sup>[2](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/banalg.pdf)</sup> |
| Spectral radius | r(a) = max{|λ| : λ ∈ σ(a)}, and Gelfand's formula gives r(a) = lim ‖a<sup>n</sup>‖<sup>1/n</sup><sup> • </sup><sup>[3](https://encyclopediaofmath.org/wiki/Banach_algebra)</sup> |
| Dependence on the algebra | If B is a closed subalgebra of A containing the unit, then σ<sub>B</sub>(a) ⊂ σ<sub>A</sub>(a); the spectrum can shrink when the ambient algebra shrinks<sup>[1](https://www.math.ucsd.edu/~bdriver/241B_W2020/Lecture%20Notes/241Functional_2020_Ver3.pdf)</sup> |
| Self-adjoint operators | A bounded self-adjoint operator on a Hilbert space has real spectrum contained in [−‖A‖, ‖A‖], with empty residual spectrum<sup>[4](https://www.math.ntu.edu.tw/~chern/notes/ApplAnalysis2018.pdf)</sup> |
| Compact operators | For a compact operator on a Hilbert space, the only possible continuous spectrum is 0, and nonzero eigenvalues accumulate only at 0<sup>[4](https://www.math.ntu.edu.tw/~chern/notes/ApplAnalysis2018.pdf)</sup> |
| Unbounded operators | A non-closed operator has spectrum equal to all of C; even closed unbounded operators can have empty resolvent set<sup>[5](https://www.mat.univie.ac.at/~gue/lehre/23AFA/AFA.pdf)</sup> |

## Definition in a Banach algebra

A Banach algebra is a Banach space over C equipped with an associative product satisfying ‖xy‖ ≤ ‖x‖ ‖y‖; it is unital if it contains an element 1 with 1x = x1 = x and ‖1‖ = 1<sup>[2](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/banalg.pdf)</sup>. For a ∈ A, the spectrum and resolvent set are

\[ \sigma(a) = \{\lambda \in \mathbb{C} : a - \lambda 1 \text{ is not invertible}\}, \qquad \rho(a) = \mathbb{C} \setminus \sigma(a). \]

For λ ∈ ρ(a), the resolvent operator is R(λ) = (λ1 − a)<sup>−1</sup><sup> • </sup><sup>[1](https://www.math.ucsd.edu/~bdriver/241B_W2020/Lecture%20Notes/241Functional_2020_Ver3.pdf)</sup>. The starting point of the theory is the geometric series: if ‖y‖ < 1 then 1 − y is invertible with inverse \( \sum_{k=0}^{\infty} y^k \)<sup>[2](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/banalg.pdf)</sup>. This immediately shows that λ ∉ σ(a) whenever |λ| > ‖a‖, so the spectrum is bounded.

## Basic properties

**Openness and holomorphy.** If λ₀ ∈ ρ(a), then λ − a is invertible for all λ with |λ − λ₀| ≤ 1/‖R(λ₀)‖, by writing λ − a = (λ₀ − a)(1 + (λ − λ₀)R(λ₀)) and expanding. Hence the resolvent set is open, the spectrum is closed, and λ ↦ (λ − a)<sup>−1</sup> is holomorphic on ρ(a)<sup>[2](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/banalg.pdf)</sup>. The same holds for closed unbounded operators, where the resolvent map is analytic and satisfies the resolvent equation R(λ) − R(μ) = (μ − λ)R(λ)R(μ)<sup>[6](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2024/MAT520_Lecture_Notes.pdf)</sup>.

**Nonemptiness.** If σ(a) were empty, the resolvent would be an entire function, and the bound ‖(λ − a)<sup>−1</sup>‖ ≤ C/|λ| for large |λ| would contradict Liouville's theorem. Therefore σ(a) ≠ ∅<sup>[2](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/banalg.pdf)</sup>. This is the first point where complex analysis enters the subject in an essential way, and it has no finite-dimensional analog of the same proof: in finite dimensions nonemptiness follows from the fundamental theorem of algebra.

**Spectral radius.** The spectral radius is r(a) = max{|λ| : λ ∈ σ(a)}. It always satisfies r(a) ≤ ‖a‖, and Gelfand's formula sharpens this:

\[ r(a) = \lim_{n \to \infty} \lVert a^n \rVert^{1/n}, \]

with the limit always existing<sup>[3](https://encyclopediaofmath.org/wiki/Banach_algebra)</sup>. The identities r(a<sup>k</sup>) = r(a)<sup>k</sup>, r(λa) = |λ| r(a), and r(a) ≤ ‖a‖ hold in any Banach algebra<sup>[3](https://encyclopediaofmath.org/wiki/Banach_algebra)</sup>.

**Gelfand-Mazur theorem.** A commutative, unital normed algebra in which every nonzero element is invertible is isometrically isomorphic to C; equivalently, C is the only complex normed algebra that is a field<sup>[7](https://rdmc.nottingham.ac.uk/bitstream/handle/internal/257/Functional%20Analysis/CBAnotes.pdf)</sup>. In a commutative unital Banach algebra, the spectrum of an element is exactly the set of values φ(a) taken by the characters, that is, the nonzero multiplicative linear functionals φ : A → C<sup>[8](https://apps.math.msu.edu/PageSpace/pb/jeffrey-schenker/Teaching/921_Lecture_Notes.pdf)</sup>. This identification underlies the [Gelfand representation](https://www.edgepedia.org/gelfand-representation), which realizes a commutative Banach algebra with trivial radical as an algebra of continuous functions on a compact [Hausdorff space](https://www.edgepedia.org/hausdorff-space)<sup>[8](https://apps.math.msu.edu/PageSpace/pb/jeffrey-schenker/Teaching/921_Lecture_Notes.pdf)</sup>.

## Dependence on the ambient algebra

The spectrum is not an intrinsic property of the element alone. If B is a Banach subalgebra of A containing the same unit, then an element a ∈ B invertible in A need not be invertible in B, and one always has σ<sub>B</sub>(a) ⊂ σ<sub>A</sub>(a)<sup>[1](https://www.math.ucsd.edu/~bdriver/241B_W2020/Lecture%20Notes/241Functional_2020_Ver3.pdf)</sup>. Equality holds for all a precisely when every element of A that is invertible in A and lies in B is already invertible in B<sup>[1](https://www.math.ucsd.edu/~bdriver/241B_W2020/Lecture%20Notes/241Functional_2020_Ver3.pdf)</sup>.

In the [C*-algebra](https://www.edgepedia.org/c-algebra) setting this ambiguity disappears: if C is a unital C*-algebra and A ⊂ C is a closed subalgebra containing the unit and closed under the involution x ↦ x*, then x is invertible in C if and only if it is invertible in A, so the spectra agree<sup>[2](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/banalg.pdf)</sup>. For a self-adjoint element a = a* of a unital C*-algebra, the spectral radius equals the norm, r(a) = ‖a‖<sup>[2](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/banalg.pdf)</sup>.

## Spectra of operators

For a bounded operator T on a complex Banach space X, the spectrum of T as an element of the Banach algebra B(X) coincides with the operator-theoretic spectrum, the set of λ for which T − λI lacks a bounded inverse<sup>[9](https://encyclopediaofmath.org/wiki/Spectrum_of_an_operator)</sup>. In infinite dimensions the spectrum can be much larger than the point spectrum (the set of eigenvalues): the unilateral shift on ℓ² has no eigenvalues, yet its spectrum is the closed unit disk.

The spectrum splits into three disjoint parts: the point spectrum, where ker(λ − T) ≠ {0}; the continuous spectrum, where λ − T is injective with dense but non-surjective range; and the residual spectrum, where the range fails to be dense<sup>[10](https://www.math.utoronto.ca/ivrii/PDE-textbook/Chapter13/S13.4.html)</sup>.

**Self-adjoint operators.** For a bounded self-adjoint operator A on a [Hilbert space](https://www.edgepedia.org/hilbert-space), eigenvalues are real, eigenvectors for distinct eigenvalues are orthogonal, σ(A) ⊂ [−‖A‖, ‖A‖], and the residual spectrum is empty<sup>[4](https://www.math.ntu.edu.tw/~chern/notes/ApplAnalysis2018.pdf)</sup>. Moreover r(A) = ‖A‖<sup>[4](https://www.math.ntu.edu.tw/~chern/notes/ApplAnalysis2018.pdf)</sup>. For a symmetric densely defined operator, self-adjointness is equivalent to σ(T) ⊂ R<sup>[5](https://www.mat.univie.ac.at/~gue/lehre/23AFA/AFA.pdf)</sup>.

**Compact operators.** Compact operators, the limits of finite-rank operators, behave most like matrices. For a compact operator on a Hilbert space, the only possible continuous spectrum is 0, and the only possible accumulation point of eigenvalues is 0<sup>[4](https://www.math.ntu.edu.tw/~chern/notes/ApplAnalysis2018.pdf)</sup>. A compact self-adjoint operator admits a spectral decomposition A = Σ λ<sub>n</sub>P<sub>n</sub>, where the nonzero eigenvalues are finite or countably infinite with finite-dimensional eigenspaces, ordered so |λ<sub>n</sub>| decreases to 0, and P<sub>n</sub> are the orthogonal projections onto the eigenspaces<sup>[4](https://www.math.ntu.edu.tw/~chern/notes/ApplAnalysis2018.pdf)</sup>.

**Unbounded operators.** For a closed densely defined operator T with domain D(T), the resolvent set consists of λ for which λ − T maps D(T) bijectively onto H with bounded inverse, and σ(T) = C \ ρ(T)<sup>[6](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2024/MAT520_Lecture_Notes.pdf)</sup>. Two features distinguish the unbounded case. First, a non-closed operator automatically has σ(T) = C, since invertibility of λ − T forces closedness; this is why spectra are usually defined only for closed operators<sup>[5](https://www.mat.univie.ac.at/~gue/lehre/23AFA/AFA.pdf)</sup>. Second, the spectrum depends on the domain of definition, not just the formal expression: the differentiation operator ψ ↦ −iψ′ on [0, 1] with maximal domain has σ = C, while the same expression on the domain {φ : φ(0) = 0} has empty spectrum<sup>[6](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2024/MAT520_Lecture_Notes.pdf)</sup>. Even for closed unbounded operators the spectrum need not be compact and can be empty<sup>[5](https://www.mat.univie.ac.at/~gue/lehre/23AFA/AFA.pdf)</sup>.

## Functional calculus and the spectral mapping theorem

The spectrum is the support of functional calculus. For a unital Banach algebra and f holomorphic in a neighborhood of σ(a), the [holomorphic functional calculus](https://www.edgepedia.org/holomorphic-functional-calculus) defines

\[ f(a) = \frac{1}{2\pi i} \int_{\partial V} f(z)(z - a)^{-1}\, dz, \]

where V is a bounded open set containing σ(a); the result is independent of the choice of contour, the map f ↦ f(a) is a homomorphism, and it agrees with the natural definitions for polynomials and convergent power series<sup>[11](https://oshalit.net.technion.ac.il/2014/01/05/advanced-analysis-notes-19-the-holomorphic-functional-calculus-ii-definition-and-basic-properties/)</sup>. Its centerpiece is the spectral mapping theorem: σ(f(a)) = f(σ(a)) for every f holomorphic near σ(a)<sup>[11](https://oshalit.net.technion.ac.il/2014/01/05/advanced-analysis-notes-19-the-holomorphic-functional-calculus-ii-definition-and-basic-properties/)</sup>. For normal operators on a Hilbert space, the continuous functional calculus assigns to each f ∈ C(σ(A)) an operator with σ(f(A)) = f(σ(A))<sup>[12](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf)</sup>.

## Pseudospectra

For nonnormal operators the spectrum alone can be a poor guide to behavior. If A is far from normal, meaning AA* ≠ A*A, its eigenvalues may have little to do with quantities such as ‖A<sup>n</sup>‖ or ‖exp(tA)‖; more information is carried by the pseudospectra, the level sets of the resolvent norm ‖(zI − A)<sup>−1</sup>‖ in the complex plane<sup>[13](https://epubs.siam.org/doi/abs/10.1137/S0036144595295284)</sup>. Large resolvent norms outside the spectrum make the numerical computation of eigenvalues of non-self-adjoint operators difficult, a phenomenon relevant in quantum mechanics, random matrix theory, and fluid dynamics<sup>[14](https://arxiv.org/abs/0705.4561)</sup>.

## History

The term spectrum was coined by [David Hilbert](https://www.edgepedia.org/david-hilbert) in his work on integral equations, published in six papers in the Göttinger Nachrichten from 1904 to 1910 and collected in the 1912 *Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen*<sup>[15](https://mathshistory.st-andrews.ac.uk/DSB/Hilbert.pdf)</sup>. Hilbert turned from the inhomogeneous integral equations studied by Ivar Fredholm to the homogeneous ones, working with quadratic forms on the space of square-summable sequences, and handled the continuous spectrum by means of Stieltjes integrals<sup>[15](https://mathshistory.st-andrews.ac.uk/DSB/Hilbert.pdf)</sup>. The choice of the word was, in the later judgment of the biographer, prophetic: twenty years later physicists used spectra of operators, as studied by Hilbert, to explain optical spectra<sup>[15](https://mathshistory.st-andrews.ac.uk/DSB/Hilbert.pdf)</sup>. One account records that Hilbert seems to have adopted the term from an 1897 article by [Wilhelm Wirtinger](https://www.edgepedia.org/wilhelm-wirtinger)<sup>[16](https://eclass.uoa.gr/modules/document/file.php/MATH122/History%20of%20Operator%20Theory.pdf)</sup>. Hilbert's approach to spectral resolution was simplified essentially by [Frigyes Riesz](https://www.edgepedia.org/frigyes-riesz) in 1913, and the theory was extended to unbounded self-adjoint operators by John von Neumann and Marshall H. Stone around 1930<sup>[15](https://mathshistory.st-andrews.ac.uk/DSB/Hilbert.pdf)</sup>. The abstract Banach-algebra formulation, with the spectrum of a single element at its center, was developed later, principally by Israel Gelfand, whose name attaches to the spectral radius formula and the Gelfand-Mazur theorem<sup>[3](https://encyclopediaofmath.org/wiki/Banach_algebra)</sup>.

## References

1. [Bruce K. Driver, Functional Analysis lecture notes, UCSD (Definition 3.1, spectrum of a single element)](https://www.math.ucsd.edu/~bdriver/241B_W2020/Lecture%20Notes/241Functional_2020_Ver3.pdf)
2. [Michael Taylor, Lectures on Banach Algebras, University of North Carolina](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/banalg.pdf)
3. [Banach algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Banach_algebra)
4. [Applied Analysis notes, Chapter 8: Spectrum of bounded operators, National Taiwan University](https://www.math.ntu.edu.tw/~chern/notes/ApplAnalysis2018.pdf)
5. [Gert Pedersen style lecture notes on advanced functional analysis, University of Vienna](https://www.mat.univie.ac.at/~gue/lehre/23AFA/AFA.pdf)
6. [Princeton University MAT520 Functional Analysis Lecture Notes (Fall 2024)](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2024/MAT520_Lecture_Notes.pdf)
7. [Resolvent and spectrum, Functional Analysis notes, University of Nottingham](https://rdmc.nottingham.ac.uk/bitstream/handle/internal/257/Functional%20Analysis/CBAnotes.pdf)
8. [Jeffrey Schenker, Functional Analysis Lecture Notes, Michigan State University](https://apps.math.msu.edu/PageSpace/pb/jeffrey-schenker/Teaching/921_Lecture_Notes.pdf)
9. [Spectrum of an operator, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Spectrum_of_an_operator)
10. [Michael Ivrii, PDE textbook, Section 13.4: Spectrum, University of Toronto](https://www.math.utoronto.ca/ivrii/PDE-textbook/Chapter13/S13.4.html)
11. [Orr Moshe Shalit, Advanced Analysis Notes 19: The holomorphic functional calculus II, Technion](https://oshalit.net.technion.ac.il/2014/01/05/advanced-analysis-notes-19-the-holomorphic-functional-calculus-ii-definition-and-basic-properties/)
12. [Dietmar Salamon, Functional analysis notes, Chapter 5: Spectral theory, ETH Zürich](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf)
13. [Lloyd N. Trefethen, Pseudospectra of Linear Operators, SIAM Review 39 (1997), 383–406](https://epubs.siam.org/doi/abs/10.1137/S0036144595295284)
14. [Pseudospectra of Systems of Semiclassical Operators, arXiv:0705.4561](https://arxiv.org/abs/0705.4561)
15. [David Hilbert, Dictionary of Scientific Biography, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/DSB/Hilbert.pdf)
16. [History of Operator Theory, University of Athens](https://eclass.uoa.gr/modules/document/file.php/MATH122/History%20of%20Operator%20Theory.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Operator algebras › Banach and normed algebras › Spectrum and functional calculus*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026; Oct 25, 2026 · Last review: Oct 15, 2026*

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