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The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid , in an infinite-dimensional setting.
Linear Operators is a three-volume textbook on the theory of linear operators, written by Nelson Dunford and Jacob T. Schwartz. The three volumes are (I) General Theory; (II) Spectral Theory, Self Adjoint Operators in Hilbert Space; and (III) Spectral Operators. The first volume was published in 1958, the second in 1963, and the third in 1971.
Holonomic brain theory is a branch of neuroscience investigating the idea that consciousness is formed by quantum effects in or between brain cells. Holonomic refers to representations in a Hilbert phase space defined by both spectral and space-time coordinates. [ 1 ]
Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly ...
In the theory of ordinary differential equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in ordinary ...
A rigged Hilbert space is a pair (H, Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map:, is continuous. [ 4 ] [ 5 ] Identifying H with its dual space H * , the adjoint to i is the map i ∗ : H = H ∗ → Φ ∗ . {\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}
In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation . In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space .
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.