Operator theory

Operator theory studies operators, especially linear maps on vector spaces and function spaces. It examines their algebraic, analytic, and spectral properties.

Linear operator: A map between vector spaces that preserves vector addition and scalar multiplication. Linearity is the central structure studied by operator theory.
Spectrum (functional analysis): The set of scalars for which an operator minus that scalar times the identity is not invertible. The spectrum generalizes eigenvalues and captures an operator's central analytic behavior.
Differential operator: An operator built from differentiation, often with coefficients that vary across the domain. Differential equations become operator equations when derivatives act on unknown functions.
Matrix theory: The study of matrices and their algebraic, geometric, and computational properties. Finite-dimensional matrices are familiar models, while operator theory also handles infinite-dimensional spaces.
David Hilbert: A German mathematician whose work helped establish functional analysis and the theory of integral equations. Hilbert spaces and integral-equation methods became central settings for operator theory.
Banach space: A complete normed vector space. Completeness lets analysts study limits of operator sequences and series.
Resolvent operator: The inverse of an operator minus a scalar multiple of the identity, where that inverse exists. Its dependence on the scalar encodes spectral information.
Integral operator: An operator that maps a function to an integral involving a kernel and that function. Integral equations and many transforms are naturally studied as operators.
Unbounded operator: A linear operator that is not bounded on its domain, often defined only on a dense subspace. Many differential operators are unbounded, requiring domain-sensitive methods beyond bounded operator theory.
Stefan Banach: A Polish mathematician who founded modern functional analysis and proved foundational theorems about Banach spaces. His work supplied the complete normed spaces on which much operator theory is built.

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