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Extra resources for A problem in the spectral theory of an ordinary differential operator in a complex domain

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Proof. ] stating that the complementation in (T J ) is (=J)_ invariant. 14. We shall study the following situation. There is given a tribe (T) and an ideal J in it. In addition to that we have another tribe (5) which is a finitely genuine supertribe of (T). (See def. [A. ) Theorem. *) We shall prove that then the tribe (T J) is a finitely genuine subtribe of (SId. 14a. Proof. The somata of (T J) are somata of (T), and conversely. First we shall prove that, for somata a, b, c, ... of (T J)' we have the equivalence of the statements 1)a+ J b=J c , 2) a +K b =K c.

CoFn). + + + p(F}. ], we prove the statement. The above lemmas yield proofs for the following properties of the distance of somata of (G). 11. 12. J. lEI + F21 ~ lEI, E21 + ... + En, Fl + ... ]. 13. ]. 14. 15. Proof by + IFl' F21· lEI, Fll + ... + + IFl' F 21. lEI E2 ... En, Fl F 2 . ]. + ... + lEn Fnl. IE, FI = IcoE, coFI. ]. 1 We shall optionally write result. IE, FI instead of IE, FI", when no ambiguity can A. 16. ]. 17. IE H, F HI ~ IE, Fl. Proof. ], the statement follows. tS. IE + F, HI s IE, HI + IF, HI· Proof.

The above shows that the notion of distance IE, F II' organizes (G) into a Hausdorff-metric topology, in which however not (=) is the governing equality, but (="). Indeed the relation IE, FI I , = 0 is equivalent to E =fl F. 4. e. if E =1' F, then fl(E) = fl(F). E, all' = IF, 011 hence fl(E) = fl(F). 20. The set I of all somata E with fl (E) = is an ideal in the tribe (G). ]. Proof. If E 1 ,E2.... ,E" .... EI, then fl(E 1 +E2 +···)=O, and then EI + E2 + ... E I. On the other hand we have: if E EJ and E' < E, then E' E I.

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A problem in the spectral theory of an ordinary differential operator in a complex domain by V. A. Tkachenko


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