Lomonosov's invariant subspace theorem
Lomonosov's invariant subspace theorem is a mathematical theorem from functional analysis concerning the existence of invariant subspaces of a linear operator on some complex Banach space. The theorem was proved in 1973 by the Russian–American mathematician Victor Lomonosov.[1]
Lomonosov's invariant subspace theorem
Notation and terminology
Let [math]\displaystyle{ \mathcal{B}(X):=\mathcal{B}(X,X) }[/math] be the space of bounded linear operators from some space [math]\displaystyle{ X }[/math] to itself. For an operator [math]\displaystyle{ T\in\mathcal{B}(X) }[/math] we call a closed subspace [math]\displaystyle{ M\subset X,\;M\neq \{0\} }[/math] an invariant subspace if [math]\displaystyle{ T(M)\subset M }[/math], i.e. [math]\displaystyle{ Tx\in M }[/math] for every [math]\displaystyle{ x\in M }[/math].
Theorem
Let [math]\displaystyle{ X }[/math] be an infinite dimensional complex Banach space, [math]\displaystyle{ T\in\mathcal{B}(X) }[/math] be compact and such that [math]\displaystyle{ T\neq 0 }[/math]. Further let [math]\displaystyle{ S\in\mathcal{B}(X) }[/math] be an operator that commutes with [math]\displaystyle{ T }[/math]. Then there exist an invariant subspace [math]\displaystyle{ M }[/math] of the operator [math]\displaystyle{ S }[/math], i.e. [math]\displaystyle{ S(M)\subset M }[/math].[2]
Citations
- ↑ Lomonosov, Victor I. (1973). "Invariant subspaces for the family of operators which commute with a completely continuous operator". Functional Analysis and Its Applications 7: 213–214.
- ↑ Rudin, Walter. Functional Analysis. McGraw-Hill Science/Engineering/Math. p. 269-270. ISBN 978-0070542365.
References
- Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi.
category:Functional analysis category:Operator theory
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