Fie \( X \) un spatiu normat si \( A\in\mathcal{L}_{C}(X) \) (operator compact), \( A\neq O_{\mathcal{L}(X)} \). Atunci pentru orice \( B\in\mathcal{L}(X) \) care nu este scalar si comuta cu \( A \), exista \( Y\subset{X} \) propriu, \( Y \) inchis, invariant la orice \( \delta\in\mathcal{L}(X) \) care comuta cu \( B \).
Last edited by Cezar Lupu on Sun Jun 08, 2008 7:37 pm, edited 1 time in total.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
Teorema spune de fapt ca orice operator care comuta cu un operator compact are subspatii invariante si este adevarata pentru spatii Banach infinit-dimensionale. Demonstratia se gaseste in articolul lui Lomonosov, Invariant subspaces for operators commuting with compact operator, Funkcional Anal. i Priloien, 7:3(1973), pag 55-56 sau in V. Rundle - Linear Analysis si foloseste in mod esential teorema de punct fix a lui Schauder.
