By William K. Allard, Frederick J., Jr. Almgren

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C H2 is invariant for 5* and T* = S*\J%*. The proposition obviously follows now from Beurling's classification of invariant subspaces for S. Indeed, either %f = H2 or %? =^ H2 and H2 Q%? = 9H2 for some inner function 9. In this last case, & = &(0) and T = S{9). 7. COROLLARY. For every inner function 9 the adjoint S(9)* is unitarily equivalent to S(9~). PROOF. 6 and T is not a shift because it is of class Coo- Therefore T is unitarily equivalent to S(9') for some inner function 9'. 1; the corollary follows.

3, rnr ^ m ^ m r " 2. For certain operators T of class Co the only hyperinvariant subspaces have the form ker#(T); a trivial example is T = 0. Is it always true that every hyperinvariant subspace of T has the form ker0(T)? 3. (<%*) be an operator of class Co- Show that there exist hyperinvariant subspaces &' and %f" for T such that XT1 n#"' = {0}, ^'V^ff = XT, ™>T\<%" is a Blaschke product, and mT\^n is a singular inner function. 4. " in the preceding problem are uniquely determined. 5. " be as in Exercise 3.

4 we have a(T | ker0(T)) C supp(0) C A. 19. /#) = 0 and we conclude that JK C ker0(T). The proposition follows. 4 . 1 3 . PROPOSITION. {%f) be an operator of class C0 and let { G i , G 2 , . . , G n } be a finite open covering of a(T). For each j , 1 < j < n, choose a localization 0j of TUT to Gj. Then we have ker 0i (T) + ker 02(T) + • • • + ker0 n (T) = W. PROOF. , Dn} be an open covering of a(T) such that Dj C Gj, 1 < j < n. 14) iirf{|(mr/0j)(*)l: A € Dj n D } > 0, 1 < j < n. 15) covers the support of mr- Therefore inf||(mT/^)(A)|:AeD\mAjl>0, 1 < j < n.

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