Operator theory is the branch of functional analysis that studies linear operators on function spaces, beginning with the differential and integral operators that motivated the field. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionGiven a linear operator on a Hilbert or Banach space, what can be said about its spectrum, its invariant subspaces and its classification, considered one operator at a time before any attempt to organize operators into families. 1 Key DebateWhether every bounded operator on a complex Banach space has a proper closed invariant subspace, the invariant subspace problem, which remains unresolved despite the field's use of the continuous functional calculus and von Neumann algebras to settle many related, more specific questions. 2 Classification
Pure or Applied Connections
Associated With
Includes
Source MacTutor biography of Alain Connes, St Andrews
Source Choi's theorem on completely positive maps - Wikipedia
Source Connes Embedding Problem, Wikipedia
Source Wolfram MathWorld: Involution
Source Erik Ivar Fredholm (Wikipedia)
Source Jacques Dixmier (MacTutor biography)
Source MacTutor biography of Mark Krein, St Andrews
Source Wolfram MathWorld: Operator
Sources
1. Operator Theory (Wikipedia)
WikipediaLead/Introduction
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators.
Single operator theory
Single operator theory deals with the properties and classification of operators, considered one at a time.
Polar decomposition
By property of the continuous functional calculus, |A| is in the C*-algebra generated by A. A similar but weaker statement holds for the partial isometry: the polar part U is in the von Neumann algebra generated by A.
View the Source 2. Invariant Subspace Problem (Wikipedia)
WikipediaIntroduction/LeadQuote, Introduction/Lead
the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself.
Choi's theorem on completely positive maps - Wikipedia
Includes: Choi's Theorem on Completely Positive Maps, LeadQuote, Includes: Choi's Theorem on Completely Positive Maps, Lead
Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras
Jacques Dixmier (MacTutor biography)
Includes: Jacques Dixmier, SummaryQuote, Includes: Jacques Dixmier, Summary
Jacques Dixmier is a French mathematician who worked on C*-algebras and other operator algebras.
Erik Ivar Fredholm (Wikipedia)
Includes: Ivar Fredholm, Lead paragraphQuote, Includes: Ivar Fredholm, Lead paragraph
operator theory
Wolfram MathWorld: Involution
Wolfram MathWorldIncludes: Involution (Mathematics), Subject classificationsQuote, Includes: Involution (Mathematics), Subject classifications
Calculus and Analysis Operator Theory
Wolfram MathWorld: Operator
Wolfram MathWorldIncludes: Operator (Mathematics), Subject classificationsQuote, Includes: Operator (Mathematics), Subject classifications
Calculus and Analysis Operator Theory
Connes Embedding Problem, Wikipedia
Includes: Connes Embedding Problem, Lead sectionQuote, Includes: Connes Embedding Problem, Lead section
by Alain Connes in the 1970s, is a major problem in von Neumann algebra theory. During that time, the problem was reformulated
MacTutor biography of Alain Connes, St Andrews
Includes: Alain Connes, SummaryQuote, Includes: Alain Connes, Summary
Alain Connes is a French mathematician and theoretical physicist who works in operator algebras and noncommutative geometry.
MacTutor biography of Mark Krein, St Andrews
Includes: Mark Krein, SummaryQuote, Includes: Mark Krein, Summary
Mark G Krein was a Soviet mathematician who worked in functional analysis, operator theory, classical analysis and representation theory.
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