Branches of Mathematics
Operator Theory
Citation Formats
General Reference
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BibTeX
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.
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 Cross-Tradition Connections
Sources
1. Operator Theory (Wikipedia)
WikipediaLead/IntroductionQuote, Lead/Introduction
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators.
1. Operator Theory (Wikipedia)
WikipediaSingle operator theoryQuote, Single operator theory
Single operator theory deals with the properties and classification of operators, considered one at a time.
1. Operator Theory (Wikipedia)
WikipediaPolar decompositionQuote, 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.
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.
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