In functional analysis and operator algebra, spectral theory is the theory of spectra of linear operators.
These operator spectra generalize the sets of eigenvalues of matrices on finite-dimensional vector spaces to the case of infinite-dimensional topological vector spaces, where eigenvalues of operators are accompanied by more general and more richly behaved values for which the combination does not have a bounded linear inverse.
Spectral theory has many applications, notably to quantum mechanics, but also for instance as the foundation of the Gelfand-Neumark theorem.
Loosely related is the algebraic geometry of spectra of commutative rings and its categorification to spectra of abelian categories (cf. the spectral cookbook).
spectral flow in geometry; related to study of index theorems
In Gnang, Elgammal, Retakh 2010 some analogues of spectral theorems are proved for tensors of higher rank, rather than just for matrices.
spectrum of an operator, spectra of families, of a Banach algebra, Gelfand spectrum, prime (primitive etc.) spectrum of a commutative or noncommutative ring, analytic spectrum, spectrum of an abelian category, spectrum of a triangulated category, Berkovich spectrum, Orlov spectrum etc.
in algebraic geometry, especially in birational geometry and arithmetic geometry, points are sometimes reconstructed by looking at valuations on the field of rational functions.
Jean Dieudonné, Fundamentals of modern analysis, vol. I, chapter XI: Elementary spectral theory (1969)
Jean Dieudonné, Sur la théorie spectrale, J. Math. Pures Appl. (9) 35 (1956), 175–187, MR0077894
Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Academic Press (1978) Volume IV: Analysis of Operators [ISBN:9780080570457]
William Arveson, A Short Course on Spectral Theory, Graduate Texts in Mathematics 209, Springer (2002) [doi:10.1007/b97227]
Nelson Dunford, Jacob T Schwartz, Linear operators, spectral theory, self adjoint operators in Hilbert space (Part 2) (1967, paperback 1988). Wiley.
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, (1995) [gBooks]
Vladimir Müller: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Operator Theory: Advances and Applications 139, Birkhäuser (2003, 2007) [doi:10.1007/978-3-7643-8265-0]
Valter Moretti, Spectral Theory and Quantum Mechanics – Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, Springer (2017) [doi:10.1007/978-3-319-70706-8]
See also:
eom: Spectral theory of linear operators (by Hazewinkel)
Wikipedia: Spectral theory
Tom Leinster, Spectra of operators and rings, nCafé
A. L. Rosenberg, The spectrum of abelian categories and reconstructions of schemes, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. 197, Marcel Dekker, New York, 257–274, 1998; MR99d:18011; and Max Planck Bonn preprint Reconstruction of Schemes, MPIM1996-108 (1996)
A. L. Rosenberg, Spectra of noncommutative spaces, MPIM2003-110 ps dvi (2003);
Underlying spaces of noncommutative schemes, MPIM2003-111, dvi, ps
Last revised on November 13, 2025 at 08:42:28. See the history of this page for a list of all contributions to it.