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Operator Theory

Analysis

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 Question
Given 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 Debate
Whether 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
Pure Mathematics 1
Operator Theory
Filter Results1 entry

Gleason's Theorem

Operator Theory
Connections

Associated With

Functional Analysis, Branches of Mathematics

Includes

David Hilbert, Mathematicians
Gleason's Theorem, Theorems
Ivar Fredholm, Mathematicians
Source Erik Ivar Fredholm (Wikipedia)
John von Neumann, Mathematicians
Sources
1. Operator Theory (Wikipedia)
Wikipedia
  • Lead/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.
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2. Invariant Subspace Problem (Wikipedia)
WikipediaIntroduction/Lead
Quote, 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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Erik Ivar Fredholm (Wikipedia)
Includes: Ivar Fredholm, Lead paragraph
Quote, Includes: Ivar Fredholm, Lead paragraph
operator theory
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