47: Operator theory

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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 47: Operator theory

Introduction

Operator theory studies transformations between the vector spaces studied in Functional Analysis, such as differential operators or self-adjoint operators. The analysis might study the spectrum of an individual operator or the semigroup structure of a collection of them.

History

See e.g. Felippa, Carlos A.: "50 year classic reprint: an appreciation of R. Courant's "Variational methods for the solution of problems of equilibrium and vibrations" [Bull. Amer. Math. Soc. 49 (1943), 1--23; MR 4, 200]", Internat. J. Numer. Methods Engrg. 37 (1994), no. 13, 2159--2187.

Applications and related fields

[Schematic of subareas and related areas]

Subfields

47A: General theory of linear operators
47B: Special classes of linear operators
47C: Individual linear operators as elements of algebraic systems
47D: Groups and semigroups of linear operators, their generalizations and applications. See also 46LXX
47E05: Ordinary differential operators, See also 34BXX, 34LXX, 58F19
47F05: Partial differential operators, See also 35PXX, 58G05
47G: Integral, integro-differential, and pseudodifferential operators
47H: Nonlinear operators, For global and geometric aspects, see 58-XX, especially 58CXX
47J: Equations and inequalities involving linear operators [new in 2000]
47L: Linear spaces and algebras of operators (See also 46Lxx) [new in 2000]
47N: Miscellaneous applications of operator theory, see also 46NXX
47S: Other (nonclassical) types of operator theory, see also 46SXX

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Jörgens, Konrad: "Linear integral operators", Surveys and Reference Works in Mathematics, 7. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. 379 pp. ISBN 0-273-08523-9 MR83j:45001 (German original: MR 57#1036)

There is a journal specializing in this area: Integral Equations and Operator Theory

"Reviews on Operator Theory, 1980-1986", AMS

Software and tables

Other web sites with this focus

Here are the AMS and Goettingen resource pages for area 47.

Selected topics at this site

Norms of operators on Hilbert space.
Recognizing a tensor product of operators on Hilbert spaces
Heisenberg's Inequality on non-commuting operators
Parameterizations of unitary operators on a Hilbert space (and thus parameterization of the unitary and orthogonal groups).
What's beyond Bochner integrals? (Pettis integral, Birkhoff integral)
The Mean Ergodic Theorem -- average values of iterates of an operator on Hilbert space.

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