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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Điểm tương tác như giới hạn của sự tương tác tầm ngắn. | Copyright by INCREST 1981 J. OPERATOR THEORY 6 1981 341-349 THE INVARIANT SUBSPACES OF A VOLTERRA OPERATOR JOSÉ BARRIA Let ữi ốj . . am bm m 1 be disjoint subintervals of X 0 c c 0 such that b1 a2 a . Let be a measure on Xsuch that p is the Lebes- m gue measure on ữj bị 1 i m and p is purely atomic on A _J ữ Ố with a finite number of atoms in 0 c J z . Let Vp be the bounded linear 1 operator on L2 X p defined by x f t dp t fe L2 X p . 0 .r The purpose of this paper is to determine all the closed invariant subspaces of Vfi. For 0 a c let La L a resp. denote the closed subspace of all functions in L.fX p which vanish on 0 a 0 a resp. a.e. u . Since VflLa c L a La it follows that the subspaces La and L a are invariant under Vp. It is easily seen that La A L if and only if p a 0. Theorem a. The subspaces La and L a 0 A a A c are the only invariant subspaces of Vfl. If Ơ is the Lebesgue measure on X then the operator Fr on L2 0 c is the usual Volterra operator and in this case the assert on of the theorem is a well known result 2 4 5 . In particular the lattice of invariant subspaces of Vp has the order type of 0 1 . In 1 the theorem was proven for a measure p which is the Lebesgue measure with a finite number of atoms in 0 c . In this case the lattice of invariant subspaces of V t has the order type of the chain 0 1 u 2 3 u . u 2n 2n 1 where n is the number of atoms of p. In 3 p. Rosenthal gave an example of an operator whose lattice of invariant subspaces has the order type of the chain 0 1 u 2 3 . . n n ỳ 2 . Theorem A implies that the ordinal sum of a finite number of such chains can be realized as the lattice of invariant subspaces of an operator . 342 JOSÉ BARRIA Let Xa denote the characteristic function of the set A in X. Let x1 x2 . x be the atoms of 1 if it has atoms so arranged that 0 Xj x2 x c. By convention we take x0 0. Lemma 1. The adjoint of is given by m f t dfx t f e L2 X . x c Furthermore kerK lw0 Ấ e c u0 Z im c n x and kerF Ấí 0 Ả 6 c v0 Z 0. iln X
