![]() ![]() (1954), "Invariant subspaces of completely continuous operators", Annals of Mathematics, Second Series, 60 (2): 345–350, doi: 10. (2002), An Invitation to Operator Theory, Graduate Studies in Mathematics, vol. 50, Providence, RI: American Mathematical Society, doi: 10.1090/gsm/050, ISBN 978-0-8218-2146-6, MR 1921782 ^ See Pearcy & Shields (1974) for a review.A version of that proof, independently discovered by Aronszajn, is included at the end of that paper. ^ Von Neumann's proof was never published, as relayed in a private communication to the authors of Aronszajn & Smith (1954).Enflo's method of ("forward") "minimal vectors" is also noted in the review of this research article by Gilles Cassier in Mathematical Reviews: MR 2186363 Typically the union of two subspaces is not a subspace. e text Think: intersection of planes (through the origin) in 3d. ^ in Foiaş, Ciprian Jung, Il Bong Ko, Eungil Pearcy, Carl (2005). X is a subspace of X A subspace not equal to the entire space X is called a proper subspace If M and N are subspaces of a vector space X, then the intersection M N is also a subspace of X. Suppose S1 and S2 are subspaces of a vector space V, and define S1 + S2 to be the set of all vectors in the form s1 + s2, where s1 is in S1 and s2 is in S2.^ See, for example, Radjavi & Rosenthal (1982).Argyros & Haydon (2009) harvtxt error: no target: CITEREFArgyrosHaydon2009 ( help) gave the construction of an infinite-dimensional Banach space such that every continuous operator is the sum of a compact operator and a scalar operator, so in particular every operator has an invariant subspace.This completely solves the non-Archimedean version of this problem, posed by van Rooij and Schikhof in 1992. Śliwa (2008) proved that any infinite dimensional Banach space of countable type over a non-Archimedean field admits a bounded linear operator without a non-trivial closed invariant subspace. ![]() Atzmon (1983) gave an example of an operator without invariant subspaces on a nuclear Fréchet space. ![]()
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