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By Curtiss D.R.

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It follows n=O that the nullspace of (S* - At has dimension at most n. ( lII=k are • "",,;>2 (. , 00 I m! m- k)1. AIII - k eIII Im(m-1) ... (m-k+1)A m - k I2

Halmos [1]). Moreover, each O"k is a polynomial in (non-negative powers of) z, since the negative Fourier coefficients of 4> vanish. Thus M"k =O"k(U) for each k. Recall that 100k(z)1 ~ 14>(z)1 a. e. Chapter 3. Shift Operators 38 We claim that CFk/U) converges strongly to Mq,. For, if fE22, then II(Mq, -CFk/U)) f 112 = JI( -CFk) fl2 dJl, and, by the Lebesgue dominated convergence theorem (Halmos [1 ]), lim J1(-CFk )fI 2dJl=0. kr-+oo J Thus Mq, is the strong limit of a sequence of polynomials in U; the second assertion of the theorem follows immediately from this fact.

19. 0 A Vn(xe o) = vn A(xe o) n < O. 2. We first discuss reducing subspaces. 21. }. Proof. 19 that a subspace JIt reduces U if and only if the projection onto ",I{ has the form P for some PE fF. Then P(z) is a projection a. e. 17, and the result follows. 0 If JV is a subspace of %, we use Jf2(JV) to denote the subspace (fE Jf2(%) :f(Z}EJV a. } of Jf2(%). 22. A subspace JIt of Jf2(%) reduces S there exists a subspace JV of % such that At = Jf2(JV}. and only if Proof. Obviously each such subspace reduces S.

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